MATH LI3RAR CORNELL UNIVERSITY LIBRARY 924 062 612 084 DATE DUE (!>^^^mm. f^Stt.1 CAYUOHO i^ntNTEDlN U.« A. The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924062612084 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1992. (QnrttEU UnioerHttg ikfcrarH 3ltlfata, £?etn ^atk ..Xo.h.n y.ew.r.Li.. Tanner. gjUBBSU^TIGS M cxyt^ty^ THE THEORY OF DETERMINANTS. Mm A SHORT COURSE THEORY OF DETERMINANTS, BY LAENAS GIFFOED WELD, PROFESSOR OF MATHEMATICS IN THE STATE UNIVERSITY OF IOWA. WeiD gork: MACMILLAN AND CO. AND LONDON. 1893. [All rights reserved.] COPTRICIHT, 1893, By MACMILLAN AND CO. Norfaorjtt ^BrreB : J. S. Gushing & Co. — Berwick & Smith. Boston, Mass., U.S.A. PREFACE. The aim of the author of the present work has been to develop the Theory of Determinants in the simplest possible manner. Great care has been taken to introduce the subject in such a way that any reader having an acquaintance with the principles of elementary Algebra can intelligently follow this development from the beginning. The last two chapters must be omitted by the student who is not familiar with the Calculus, and the same is to be said in reference to some few of the preceding arti- cles ; but in no case will the continuity of the course be affected by such omissions. No attempt has been made to apply the theory to Analytical Geometry, though a few of its more im- portant applications to Algebra have been included. The reader familiar with geometrical analysis will be interested in giving to the greater number of these applications, as also to many of the examples, their geometrical interpretations. The earlier the student, in his mathematical course, is made familiar with the notation and methods of Determinants, the earlier will he be VI PREFACE. prepared to appreciate the wonderful symmetry and generality so characteristic of the various modern developments in mathematics. In consideration of the limited time available for the study of such topics in the ordinary college course, the attempt has been made to render the book as readable as possible, rather than to prepare a drill book. How- ever, it is hoped that the student who solves the two or three hundred examples proposed may thereby receive valuable mental discipline. Acknowledgments are due the writings of Muir, Scott, Hanus, Salmon, Baltzer, Gunther, Olehsch, Mansion, Dostor, Houel, and others. Even greater indebtedness to some of the above-named writers would doubtless have added much to whatever of merit the work now submitted may possess. L. G. W. Iowa Citt, Iowa, U.S.A. 1893, March 10th. CONTENTS. CHAPTER I. The Origin and Notation of Determinants. ART. PAGE 1, 2. Determinants of the second order .... 2 3. Solution of pairs of linear equations by determinants . 3 Examples 4 4, 5. Determinants of the third order 7 6. Solution of systems of three linear equations by deter- minants 9 Examples 10 7. Determinants of higher orders 14 8. Notations and definitions 16 CHAPTER II. General Definition of a Determinant. 9. Permutations. Permanences and inversions. Even and odd permutations 20 10. Interchange of two members of a group changes the character of the permutation 23 11. Number of even permutations equal to the number of odd 25 12. Definition of a determinant 25 13. Signs of terms determined by permutations of sub- scripts or superscripts, indifferently ... 27 14. The 2 notation 29 vii VIU CONTENTS. CHAPTER III. Properties of Determinants. ART. PAGE 15. Columns may be changed into rows, and rows into columns . 30 16. Interchanging two rows changes sign of determinant . 31 17. Any element may be brought into the leading position 32 18. Determinant vanishes if two rows are identical . . 32 19. Resolution of determinant having a column of poly- nomial elements . 33 20. Multiplying each element of a column by a common factor, etc. ..... . . 34 21. Elements of one row multiples of those of another . 34 22. Composition of columns and rows .... 35 23. Factoring of determinants . .... 36 Examples ......... 38 CHAPTER IV. Determinant Minors. 24. Definition of co-factor .... 25. Sign of co-factor. Notation for co-factors 26. Definition of minor .... 27. Pormulse for the expansion of determinants 28. The zero formula Examples 29. Elements on one side of the diagonal equal to 30. Raising the order of a determinant Examples 31. General definition of a minor 32. Complementary minors . 33. Product of complementary minors 34. Co-factor of a minor .... 35. Sign factor of the co-factor of a minor . 44 46 48 48 50 51 60 62 63 64 65 CONTENTS. IX ART. 36. 37. 38. 39. Notation for the co-factor of a minor Laplace's metliod of expansion Cauchy's method of expansion Examples .... Differentiation of determinants Examples .... PAGE 69 70 71 74 76 78 CHAPTER V. Applications of DETEitMiNANTS to Elementary Algebra. in n 40. General solution of systems of linear equations Examples 41. Consistence of linear systems 42. Consistence of homogeneous linear systems . 43. Relation of co-factors in a determinant which equals zero 44. System of (k — 1) homogeneous linear equations unknowns 45. The matrix of 7i columns and (m — 1) rows 46. Matrices in general .... Examples ...... 47. Sylvester's dialytic method of elimination 48. Euler's method of elimination Examples ...... 49. Discriminant of ax^ + by'^ + 2 hxy . 50. Discriminant of ax^ -|- by'' + cz^ + 2fyz + 2 gxz + 2 hxy Examples ......... 80 85 86 93 95 96 97 100 102 107 111 117 118 121 51. 52. CHAPTER VI. Multiplication of Determinants, and Reciprocal Determinants. Product of two determinants of the second order . General rule for the multiplication of determinants 124 126 X CONTENTS. ART. PAGE 53. General formula for the product of two determinants . 127 64. Linear transformation of a system of linear functions . 127 55. Product of two determinants of the nth order express- ible in (n+1) forms .... 56. Product of two vertical matrices . 57. Product of two horizontal matrices 58. Euler's theorem relating to sums of four squares Examples . 129 131 132 133 135 Beciprocal Determinants. 59. Definition of reciprocal determinant .... 138 60-63. Theorems relating to reciprocal determinants . . 139 CHAPTER VII. Determinants of Special Forms, Symmetrical Determinants. 64. Conjugate elements defined 143 65. Properties of axi-symmetric determinants . . . 144 66-68. Theorems relating to axi-symmetric determinants . 145 69. Definition of per-symmetric determinants . . . 147 70. Successive difference series formed from elements . 148 71. Theorem relating to per-symmetric determinants . 148 72. Definition of circulants 149 73, 74. Theorems relating to circulants .... 150 Skew Determinants. 75. Definition of skew and of skew-symmetric determi- nants ......... 151 76. Properties of skew-symmetric determinants . . 152 77. Skew-symmetric determinant of odd order equal to zero 153 CONTENTS. XI ART. PAGE 78. Skew-symmetric determinant of even order a perfect square 154 Pfaffians. 79. Definition of Pfaffians . . . . . 155 80. Notation of Pfaffians 157 81. Order of a Pfaffian. Frame lines . . 158 82. PfafBan minors . 159 83. Bordered slcew-symmetric determinant equal tu tlie product of two Pfaffians .... 84. Other relations to skew-symmetric determinants 85. Expansion of Pfaffians ... 86. Number of terms in the expansion E-XAMPLES 159 103 164 166 166 Alternants. 87. Alternating and symmetric functions .... 167 88. Definition of alternants. Notation . . . 168 89, 90. The difference product as an alternant . . . 169 91. Alternant whose elements are rational integral functions 171 92. Every alternant divisible by the difference product of its elements 172 93. The coefficients of the rational integral function ex- pressed as symmetrical functions of the roots . .173 94. Theorem relating to alternants derived from the preced- ing article 174 95. A determinant for the square of the difference product 175 Examples 176 Continuants. 96. Definition and simple properties of continuants . . 178 97. Expansion of a continuant in terms of co-axial minors 180 98. Number of terms in the expansion of a continuant . 180 Xll CONTENTS. ART. PAGE 99. Negative sign interchangeable between the two minor diagonals 182 100-102. llelation between convergents of descending con- tinued fractions and continuants .... 183 103. Expression of convergents of ascending continued fractions in determinant form .... 186 104. Transformation of ascending into descending continued fractions 189 105. Transformation of series into continued fractions . 190 Examples 191 CHAPTER VIII. Jacobians, Hessians, and Wkonskians. Jacobians. 106. Definition of Jacobians 107. Tlie Jacobian of a set of linear functions 108. .7i(!/i, •■-,?/„) -Jj (.1:1, •••,x„)= 1 . 109. Jacobian of indirect functions .... 110. Jacobian of implicit functions 111. Bertrand's definition of a Jacobian 112. Resolution of Jacobians into factors 113. If the Jacobian vanishes, the functions are dependent 114. The converse of the preceding .... 194 195 196 197 197 199 201 202 203 115. The Jacobian as a covariant 204 Hessians. 116. Definition of a Hessian 205 117. The Hessian as a covariant 206 K Functions. 118. Derivation of K Functions from Jacobians . . . 208 119. Second method of derivation . . . 209 CONTENTS. XIU Wronshians. ART. PAGE 120. Definition of a Wronskian 210 121,122. Formulae relating to Wronskians .... 211 123. The Wronskian vanishes if the functions have a linear relation 212 124. The converse of the preceding .... 213 Examples 214 CHAPTER IX. Linear Teansfokmations. 125. Definition of a quantio. Notation .... 219 126. Discriminants 220 127. Invariants 221 128. Covariants 221 129. The discriminant and the Hessian of a quadric are invariants 222 130. 131. Cases in which the discriminant is equal to zero . 223 132. The invariants A, 0, 0' and a' of the quadric kq-\-q' . 225 133. The orthogonal transformation 226 134. The coefficients of orthogonal substitutions of any order 228 135. Cogredient variables. Emanents .... 232 136. Contragredient variables 233 137. Reciprocal quadrics 235 Examples 237 THE THEORY OF DETERMINANTS. CHAPTER I. THE ORIGIN AND NOTATION OF DETERMINANTS. In the various processes of analysis there are cer- tain classes of functions which occur with remarkable frequency. Of these there are some which have been studied with great care, and of which general theories have been developed. The functions known as determinants constitute such a class, and one which is of great importance in some of the modern developments of mathematics. One of the simplest of the many processes giving rise to determinants, and that which led Leibnitz in 1693 to their discovery, is the solution of systems of simultaneous equations of the first degree. In the present chapter we shall give examples of a few determinants resulting from this process, and explain the general principles of the notation usually employed in representing such functions. 1 2 THEOKY OF DETERMINANTS. [chap. i. 1. Let US solve the two simultaneous linear equations, Oix + b^y = Ki, Multiplying the first equation by ^2) the second by — 6i, and adding the resulting products, we read- ily obtain ^^K,h,-KA (1) Similarly, (2) These two fractions, which express the values of X and y, have a common denominator, which is a function of the coefficients of x and y in the given simultaneous equations. This function, O162 — aJ3i, (3) is called the determinant of the coefficients «!, 61, a^, and J2, and in the notation of determinants it is commonly expressed by the symbol 02 62 (3') in which the coefficients are arranged in the same order as in the given equations. This symbol is called a determinant array. ART. 3.] OKIGLN A^D iJOTATlOX. 3 Of the determinant represented by the above array, aj, 61, a^, and bs are called elements or constitu- ents. The polynomial (3) is called the expanded form, or simply the expansion, of the determinant. Since each term of this expansion is the product of two elements, the determinant is said to be of the second order. 2. In the identical equation Ui 61 = ttiftj — 0261 the first member must be so interpreted that it shall represent the same function of ai, bi, a^, and b^ as the second member ; that is : ITie determinant array of the second order must be understood to mean that the product of the elements on the diagonal passing from the lower left-hand comer to the upper right-hand corner of the array is to be subtracted from the product of the elements o?i the other diagonal. 3. The numei-ators of the fractions in Equations (1) and (2) of Article 1 may also be written in the form of determinant arrays. Thus, by the pre- ceding article. THEOKY OF DETERMINANTS. [chap. I. kA — K^bi = "l fti K2 &2 ctiKj — ''a'^i — ai Kj . aa "2 Expressed in the notation of determinants, the values of x and y in the given simultaneous equa- tions are «1 h ai Kl K2 b, , and y = «2 «2 ffll &1 a2 h 02 &2 Note. — The numerator of the fraction expressing the value of X may be formed from tlie denominator of the same fraction by replacing a^ and a.^, the coefficients of x, by the absolute terms Kj and K.^, respectively. Similarly for y. 1. Expand 2. Evaluate EXAMPLES. X • 1 -3 5 3. Expand and reduce 4. Evaluate a b sm X sin y cos a; cosy Ans. 1 — X-. Ans. 15. Ans. sm{x—y). ART. 3.] ORIGIN AND NOTATION. 6. Evaluate 1 -2 . 3 -6 6. Expand and reduce 1 cos a cos« 1 ■ 7. Expand and reduce cos X sin y — sin X cos y 8. Expand and reduce logs; logy n m x^ Ans. log — 9. Expand and reduce 1 -1 1 — cos X Ans. 2sin^|-.'B. 0. Expand and reduce x + y x' + xy+y'' . X —y a? — xy + y'^ 11. Solve the simultaneous equations, x — 2y = 3, and 2x + 5y = 15. Solution. — Expressing the values of x and y in the notation of determinants, as in Article 3, we have 3 -2 1 3 15 1 5 -2 , and y = 2 15 1 -2 2 5 2 5 Evaluating the above arrays by Article 2 gives % = ^- = 5, and ?/ = | = 1. THEOKY OF DETERMINANTS. [chap. i. 12. Solve the simultaneous equations, 5x4-2^ = 43, x-^y = 5. 13. Solve the simultaneous equations, 2a; + 33!/ = 22, a;-2?/ = 4. 14. Solve the simultaneous equatioiis, 8a; + 7?/ = 208, 6a; = 60. 15. Solve the simultaneous equations, 1 + 1^5, 2_3^o. X y X .y 16. Deduce the values of x' and y' in terms of x and y from the equations, x = x' cos a — y' sin a, y = x' sin a + y' cos a. 17. Express, in the notation of determinants, the condition that the tw^o roots of the equation, ax^ + bx + c = 0, shall be equal. 18. Express aj : 6i : : aj ; &2 in determinant form. 19. Show that, if any element of a determinant of the second order is zero, the other element on the same diagonal may be replaced by any quantity vrhatever. 20. Show how any number may be expressed as a determinant of the second order. ART. 4.] ORIGIN AND NOTATION. 7 4. Let us now solve the three simultaneous linear equations, ajCK + b-2y+ c._z = Kj, a,jx + b^ + cz = K3. Multiplying the first equation by (h^c^ — hsp^, the second by —(^'iCj — 63(71), the tliird by (pic-i — liC^, and adding the resulting products, we readily obtain Kl&A Kl&aCo + Ko&sCl K0&1C3 + K061C9 — KJbXi X ^ ' ^^ . In a similar manner, (1) ~ aiVs — ai&3C2 + aj&aCi — O261C3 + «3^iC2 — OsW and ^l^Z'^S — O1&3K2 + 0-^31^1 — 0,AiK^ + Ct36lK2 — a3&2Kl (3) These three fractions which express the values of X, y, and z have a common denominator which is a function of the coefficients of x, y, and z in the given simultaneous equations. This function, afi^i — afiiC^ + afi^fii — a-Pi^s + is'^A — o-s^iCi, ■ ■ (4) is called the determinant of the coefficients a , 61, Cj, flz) ••• «3i ••■ ^3) find in the notation of determinants it is commonly expressed by the symbol THEORY OF DETERMINANTS. [chap. i. tti bi Ci (4') in which the coefficients are arranged in the same order as in the given equations. This symbol, like (3') in Article 1, is called a determinant array, and a„ 5j, Ci, a^, ■■■ are called the elements or constituents of the determinant. The polynomial (4), like (3) in Article 1, is called the expansion of the determinant, and since each term of this expansion is the product of three elements, the determinant is said to be of the third order. 5. In the identical equation. 6i = afi-fii — aib^c^ + a^b^Ci — a^-fi^ + a^bjCi — ajVi) the first member must be so interpreted that it shall represent the same function of a-^, Si, c^, a^, ••• as the second member. The following is one method of so interpreting the first member : Alongside the square array let the first two verti- cal ranks of elements he repeated in order ; thus. AKT. 6.] ORIGIN AND NOTATION. Now form the products of the elements lying in lines parallel to the diagonals of the original square, as shown above. The products formed from elements which lie on lines descending from left to right have the positive sign, the other products the negative sign. By this method* any determinant array of the third order may be expanded and evaluated. In practice it will not be necessary to repeat the first and second vertical ranks of elements, but only to imagine them repeated. 6. The numerators of the fractions in Equations (1), (2), and (3) of Article 4 may also be written in the form of determinant arrays. Expressed in the notation of determinants, the values of x, y, and z in the given simultaneous equations are Kl 6i Ci tti "1 Ci K2 h C2 a^ K-Z C2 i-'s h Cs , y = a, "3 C3 «1 bi Cl Ol h Cl ttj b. C2 a. h C2 a. 63 C3 as &3 C3 * This device is due to Sarrus. See Scott's Theory of Deter- minants, page 10. 10 THEOBY OF DETERMINANTS. [chap. I. and «! h Kl a^ h Kj a. h «3 «! h Cl a^ b. C2 03 63 C3 Note. — As in Article 3, it miy be observed that the numer- ator of the fraction expressing the value of x may be formed from the denominator of the same fraction by replacing a,, a^, and a^, the coefficients of x, by the absolute terms nj, ic^, and K„ respectively. Similarly for y and z. 1. Expand EXAMPLES. a^o 2/0 1 % 2/1 1 Kj 2/2 1 ,4?is. a;(,2/i + x.;A/o + x^y. — x^jj^ — X(,y.j — Xj^o- 2. Evaluate 12 3 3 12 2 3 1 Ans. 1 + 8+27 ■ G - 6 - G = 18. 3. Show that 1 X y cos X sin y sin a; cos y = cos(x + y). ART. 6.] ORIGIN AND NOTATION. 11 4. Evaluate I m . -I n — m — n 5. Evaluate 3 -3 4 . 3 2 -2 -1 2 1 6. Expand 1 cos y COSyS cos y 1 COS a COS 13 COS « 1 Ans. 0. 7. Show by expansion that «! h Cl = Cti h Cs Cli h Cs «! 02 a™ &1 6, b, Cl Ca Cs 8. Expand L n 7n n M I m Z N * If three planes form a trihedral angle, the face angles of which are a, /3, and 7, the above determinant is called the square of the sine of the solid angle in question, in analogy with 1 cos a cos a 1 12 THEORY OP DETERMINANTS. 9. Show by expansion that = — m Ci 61 tti [chap. 1. mOi h Cl = — m 7na2 b. c, ma^ h C3 10. Solve the simultaneous equations, 3x-2y = 0, x + 2y + z=12. Solution. — Expressing the values of x, y, and z in the nota- tion of determinants, as in Article 6, we have -2 -2 i -2 12 2 1 1 -2 i 3-2 1 2 1 and z = y = 1 -2 4 3 1 12 1 1 -2 J 3-2 1 2 1 1 -2 -2 3 -2 1 2 12 1 -2 i 3 -2 1 2 1 Evaluating the above arrays by Article 5 gives X = -V- = 2, 2/ = -« = 3, and z = «- = 4. ART. 6.] ORIGIN AND NOTATION. 13 11. Solve the simultaneous equations, x + 2y-Sz = 13, 3x + y + Az = 51, 12. Solve the simultaneous equations, a; = 18 -42/, y=71i-7^z, z = 10i-|x. 13. Solve the simultaneous equations, ? + ?/ + r = i93 ^ + y + ^- = 227, ^ + 2^ + ^ = 219. 3 5 7 ' 5 7 3 ' 7 3 5 14. Solve the simultaneous equations, x+y + z = 29, x + 2y+3z=62, ix + iy + iz = 10. 15. Solve the simultaneous equations, x + iy + \z=32, ^x + iy + \z = 15, ix + l-y + iz = 12. 16. Solve the simultaneous equations, 1 + 1 + 1 = 6,1 + 2 + ^=14, ? + ? + l = ll. X y z X y z x y z 17. From the equations. La = Mji = Ny, and aa + &^ + cy = C, deduce the values of «, fi, and y. A GMN 4- Ans. u = , etc. aMN+bLN+cLM 14 THEORY OP DETERMINANTS. [chap. i. 18. Resolve 63;' + 22a; + 18 ^^^ simpler fractions, af'+ea^ + llce + e ^ using determinants wherever possible. .. 1,2,' Ans. -\ h - x + 1 x + 2 X + '3 7. The solution of the four simultaneous linear equations, a^x + bii/ + CjZ + diW = kj, a^x + h^ + c^ + d^iv = (<-2, UsX + 63I/ + C32 + da?*; = K3, a^x + &4?/ + CiZ + diiv = Ki, would show that the values of x, y, z, and w are expressed by fractions having a common denomi- nator which is a function of the coefficients, a^, Sj, Ci, di, flj, •••^2, •••di. This function is a determinant of the fourth order of which the above coefficients are the elements. The four-square array represent- ing this determinant and the determinant in the expanded form are the members of the following identical equation : = afi^c^di — aJb2C^d^ — 016302^4 + afifi^d^ + afisCidi — ai64C3d2 — a^biC^di + a^biC^d^ + afifiidi — 046102^3 — aJjiCid^ + afifi^d^ -\- aii^Cidi — a^b^Cidg — aJb^Cidi -f- afi^'^'ol^s -\-ajifiidi— afiopid^ — a^^Cid^ + ajjfi^di + af)2Cidy — ajj^c^di — aJjiC^i + afi^Pfi^. ( 1 ) a, h Cl d. "2 62 C2 d^ a.^ h C3 d. "4 h C4 d. ART. 7.] OBIGIN AND NOTATION. 15 The solution of five simultaneous linear equations involving five unknown quantities V70uld give rise to a determinant of the fifth order, which in its expanded form contains one hundred twenty (120) terms. The following is the array representing this determinant, the notation being the same as hereto- fore: ai bi Ci di e^ a, 62 Cj ^2 % ttj 63 C3 dj 63 «4 bi C4 dn 64 = cii&jCsdies ± etc. (2) It is evident that the polynomials obtained in the above manner are far too cumbersome to be of any practical use in their expanded forms. The theory of determinants, however, enables us to manage not only such polynomials as the above, but also many other exceedingly complicated functions in a per- fectly simple and easy manner. To quote Professor Sylvester, the theory of determinants is "an algebra upon an algebra; a calculus which enables us to combine and foretell the results of algebraical oper- ations in the same way as algebra itself enables us to dispense with the performance of the special operations of arithmetic." Aside from the power which determinants thus possess as instruments of analysis, the functions themselves have, by reason 16 THEORY OF DETERMINANTS. [chap. 1. of their great fertility, abundantly rewarded the careful study which has been bestowed upon them by the last two generations of mathematicians. 8. In the preceding articles it has appeared that the determinant of the second order involves four elements, the determinant of the third order nine, that of the fourth order sixteen, and that of the fifth order twenty-five elements. In general, the determinant of the nth order involves n^ elements. Taking n letters, a,b,c,--- h, we may write the determinant array of the nth order thus : (1) «1 h Cl- ■Ih a^ h C2- ■ h as h C3- ■ h «» K c„- ■■ K It is customary to speak of a determinant array simply as a determinant. The horizontal ranks of elements are called rows of the determinant, and the vertical ranks are called columns. The rows are numbered from the top row AET. 8.] ORIGIN AND NOTATION. 17 downward, and the columns from the left-hand column to the right. In the above notation the number of the row to which a given element belongs is indicated by the subscript of the element, while the number of the column is indicated by the order of the letter in the alphabet. Thus, in the determinant (1) of this article the element e^ belongs to the sixth row and the fifth column. In any determinant the diagonal from the upper left-hand corner to the lower right-hand corner is called the principal diagonal, and the other is called the secondary diagonal. The terms of the expansion which are the products of the elements on these diag- onals, and it will appear in the sequence that there are such terms in the expansion of every determi- nant, are called respectively the principal term and the secondary term. The element at the upper left-hand corner of the array is called the leading element, and the place which it occupies is called the leading position. Thus, in the determinant (1), aAca ••• h„ is the principal term, «„6„_ie„_2 ••• Ai is the secondary term, and a^ is the leading element. Another notation for the determinant array of the nth. order is the following : 18 THEOKY OF DETERMINANTS. [chap. 1. a,' ai" a/" • . a,"" a,' a," • a/") a,' a," |and|a,,„|,. . . . (7), (8) respectively. 20 THEORY OF DETERMINANTS. [chap. ii. CHAPTER II. GENERAL DEFINITION OP A DETERMINANT. In Articles 2 and 5 special rules were given for the interpretation of determinant arrays of the second and third orders, but no such simple rules can be given for the interpretation of arrays of higher orders. We now proceed, therefore, to develop a general definition of the functions known as deter- minants ; that is, a general method of interpreting determinant arrays. 9. It is proved in elementary algebra that the number of permutations, or dispositions, of the mem- bers of a group of n things is 1 • 2 • 3 • • • w, or n\, or \n^ If the members of the group are the letters a b c d e •••, the order in which they have just been written is called the natural order. The natural order of the integers 1 2 3 4 5 ••• ART. 9.] GENERAL DEFINITION. 21 is, of course, the order of their magnitude, beginning with 1. In one, and in only one, of the permutations of the members of a group, the members are arranged in their natural order. In every other permutation the natural order is more or less deranged. Any two members of a group arranged in their natural order constitute a permanence. Thus, the pairs, a b a c be 12 13 23 are permanences. Any two members of a group arranged in an order which is the reverse of the natural order con- stitute an inversion. Thus, the pairs, b a c b d c 21 32 43 are inversions. The number of permanences and the number of inversions in any permutation of the members of a group may be found by comparing each member of the group vi^ith each following member. Thus, in the permutation a e d b c, the permanences are a e, ad, a b, a c, be; while the inversions are erf, e b, e c, d b, d c. 22 THEORY OF DETERMINANTS. [chap. ii. In the permutation 3 4 2 5 1 there are four permanences and six inversions.* The permutations of the members of a group are divided into two classes, the even or positive per- mutations and the odd or negative permutations. Even permutations are those which contain an even number of inversions- Odd permutations are those which contain an odd number of inversions. The permutations e a d b c, 3425 1, are even, because each contains an even number (six) of inversions ; while the permutations a e d b c, 34215, are odd, because each contains an odd number (five) of inversions. The significance of the terms positive and nega- tive, as applied to the classes of permutations, will appear later. * It may easily be shown that the sum of the number of per- manences and the number of inversions in any permutation of n things is given by the formula, n (n — 1) 2 ART. 10.] GENERAL DEFINITION. 23 10. Theorem. — If, in any permutation of the members of a group, two of the members be inter- changed, the character of the permutation is changed, either from odd to even or from even to odd. We shall first consider the case in which the two menibei's in question are adjacent. Let a and a be the two adjacent members, and represent col- lectively the members which precede an. by P and those which follow aa by M. The permutation may now be written P a a R. Now, exchanging the order aa for the order aa does not in any way affect the relations of these two members to the members of either P or R. The only change in the number of inversions is due to the substitution of the order aa for the order aa. If aa is a permanence, aa is an inver- sion, and the number of inversions is increased by one as the result of the exchange. If aa is an inversion, aa is a permanence, and the number of inversions is diminished by one. In either case, the interchange of a and « changes the number of inversions by one ; hence, the character of the permutation is changed, either from even to odd or from odd to even. If a and a be not adjacent, represent the mem- 24 THEORY OP DETEEMINANTS. [chap. ii. bers included between them collectively by Q. The permutation may now be written P a Q a R. Let q be the number of members in Q. We may bring the above permutation into the order P a a Q R by interchanging « with each of the q members of Q in succession, beginning by interchanging it with the right-hand member. We have thus made q interchanges of adjacent members, and have accord- ingly changed the class of the permutation q times. The order P aa Q R may be changed to the order P a Q a R by interchanging a with each member of (a^) in succession, beginning by interchanging it with the lefl>hand member. This requires (^q + 1) inter- changes of adjacent members ; that is, (g- + 1) changes in the class of the pei-mutation. In the course of all the interchanges of adjacent members thus made in passing from the order P a Q aR to the order P a Q a R the class of the permutation has been changed g-f (g+1) or {2q + \) times. Since q is an integer, (2q+\^ is an odd AKT. 12.] GENEKAL DEFINITION. 25 number, and the class of the permutation has been changed, either from odd to even or from even to odd. Hence the theorem. 11. Theorem. — Of all possible permutations of the members of a group, one-half are even and one- half are odd. Write down all the possible permutations. Now, let a new set of permutations be formed by fixing upon any two of the members and interchanging them in each permutation. The even permutations will thus be changed to odd, and the odd to even. That is, for every even permutation in the old set there is an odd one in the new, and vice versa. But, as is evident, the new set of permutations is the same as the old, only differently arranged. Hence, in either set there are as many even as there are odd permutations, or one-half the permu- tations are even and one-half are odd. 12. We are now prepared to give a general inter- pretation to the determinant array. This array has already been written in the general form, a/ a/' o/" ••• aV (n) (n) 26 THEORY OF DETERMINANTS. [chap. ii. Write down all the products which can be formed by taking as factors one, and only one, element from each column and each row of the array. There are n ! such products; for, since there must be in each product one element from each column, any one of the products without its subscripts may be written ^1 nil rilll .«'»), and the n subscripts corresponding to the rows of the array may be appended to the members of the above group in n\ different ways, giving n\ differ- ent products. Of these products, one half involve the even per- mutations, and the other half the odd permutations of the subscripts 1, 2, 3, •••n (Art. 11). JV^ow give to those products which involve the even permuta- tions of the subscripts the POSITIVE sign, and to those which involve the ODD permutations the NEGA- TIVE sign, and take their algebraic sum. The result is the expanded form of the determinant. [The portions of this article printed in italics read consecutively, and furnish a general rule for the expansion of determinant arrays. The reader will do well to verify the rule by applying it to the determinants in Articles 1 to 7. It should be mentioned that the method of expansion just given is of little practical value. It con- stitutes a general definition of a determinant, however, and as such will be used as a basis for the theorems relating to the prop- ART. 13.] GENERAL DEFINITION. 27 erties of determinants given in the next chapter. For this reason it must be thoroughly mastered.] 13. Let us assume any term of the expansion of the determinant written in the preceding article, as ± a A"<7,'" •••»/"', (1) in which hij •■■ I is any permutation of the subscripts 1, 2, 3, ••• n, involving any number, /u, (say), of inver- sions. Interchange any two of the elements the sub- scripts of which give rise to one of the /m inversions, and continue this process till all the inversions in the arrangement of the subscripts have disappeared. This will require a number, m (say), of interchanges of elements, m being even or odd according as fi is even or odd, and as a final result the elements will be so arranged that their subscripts are in the natural order, 1 2 3 ■ • • n. The term under consideration may now be written iai""a2'«'a «■••«,."', ... . (2) in which pqr---t is a certain permutation of the superscripts ', ", '", • • • '"'. Now, each of the m interchanges of two elements, by which the second form of the term has been obtained, has changed the class of the permutation of the superscripts from even to odd, or vice versa. 28 THEORY OF DETERMINANTS. [chap. ii. The original permutation was even, the superscripts having been arranged in the natural order.* Hence, the permutation pqr---t is even or odd according as m and fi are even or odd. The two per- mutations, hij---l And pq7----t are, therefore, of the same class, and the term under consideration will have the same sign, whether its sign be determined by its subscripts when in the form (1), or by its superscripts when in the form (2). It follows that it is immaterial whether we write the terms of the expansion of an array so that the superscripts of the elements are arianged in the natural order and consider each term as positive or negative according as the permutation of its sub- scripts is even or odd, or write the terms so that the subscripts are arranged in the natural order, and con- sider each term as positive or negative according as the permutation of its superscripts is even or odd. Note. — It may also be shown that if we write down all the products which can be formed by taking as factors one element from each column and each row of the array, the factors in each product being arranged neither with reference to the order of the subscripts nor of the superscripts, and take each product positive or negative, according as the sum of the number of inversions in the arrangement of the subscripts and the num- ber of inversions in the arrangement of the superscripts is even * The number of inversions in the natural order is zero, an even number. ART. 14.] GENERAL DEFINITION. 29 or odd, the algebraic sum of the resulting terms will be the expanded form of the determinant. This will furnish a good exercise for the student. 14. The determinants (2) and (3) of Article 8 may be written in the simpler forms 2 ±(o;o,"a;" •••«/"'), or t±{a,^'W^W" -a.'"), (1) and 2±(aj,ia,.2a,.3-"a^„), or 2 ± («i,pa.2,,a3,,-"a„,,), (2) respectively ; a notation which is suggested by the principles explained in the two preceding articles. 30 THEORY OF DETEKMINANTS. [chap. hi. CHAPTER III. PEOPEETIES OP DETERMINANTS. From the definition of a determinant contained in Articles 12 and 13 of the hist chapter, we now proceed to derive the more important theorems relating to the properties of determinants. . 15. Theorem. — The value of a determinant is t^^V>-ot changed hy changing the columns into corre- ct- i^^lsponding rows, and the rows into corresponding col- '^,'*^iumns; that is, a/tti" ■ ..a,<"' = a/ ^2' ••«„' ai'a." ■ ••a,"" Cli O2 • ••«,/ aX"- ••«„'"> o/"'a2<"' - ••«„" A determinant may be expanded either by per- muting the subscripts while the superscripts remain in the natural order, or by permuting the super- scripts while the subscripts remain in the natural order (Articles 12 and 13). Now, it is evident, since the subscripts refer to the rows and the super- scripts to the columns of the array, or vice versa, that changing from one of these methods of expan- ART. 16.] PKOPEltTIES OF DETEllMINANTS. 31 sion to the other amounts to the same thing as changing the columns of the array into correspond- ing rows, and the rows into corresponding columns. Hence the theorem. It follows that whatever theorem is demonstrated in regard to the rows of a determinant is also true in regard to the columns, and vice versa. 16. Theorem. — Interchanging any two rows (or columns') of a determinant changes the sign of the determinant ; thus. a.Ja,"a,"'---a.j"^ a,'a,"a,"'...at^ x„'a„"a„'"- ••«„"" a,'a,"a,"'-at^ 2 2 2 ' * ' 2 aJaJ'aJ".-a(-> a„'a,/"a„"-a<»' a3'a3'"a3"---a3<"' a/oi"'ai"..-a/»' Represent the given determinant by A, and the determinant obtained by interchanging any two rows of A by A'. Let ±a,'a/'a/'V •••«»'"' be any term of A. If A' be formed from A by inter- changing the rows whose subscripts are i and k (say), the corresponding term of A' will be ±OjVa/"a/'---oJ"'. This is evidently one of the terms of A, if we disregard its sign (Art. 12). Since, however, 32 THEORY OF DETERMINANTS. [chap. III. h i j k ••■ m and /* k j i ■■• ni are not permutations of the same class, the sign of this term in A must be T instead ±. Hence, for every term ± a,Ja^'a;"ar ■ (") of A', there is a term in A. Therefore A= — A', which is the theorem. 17. An element in the A;th row and the sth column may be brought into the leading position by (^ — 1) interchanges of adjacent rows and (s — 1) inter- changes of adjacent columns. Since, by the preced- ing article, each such interchange changes the sign of the determinant, the resulting determinant will be positive or negative according as (^^-s — 2), the total number of interchanges, is even or odd ; that is, according as (/c-l-s) is even or odd. 18. Theorem. — If two rows (or columns^ of a determinant are identical, the determinant is equal to zero; thus, A = = 0. ,.(") ,.(") ART. 19.] PROPERTIES OF DETERMINANTS. 33 For, by interchanging the two identical rows (Art. 16), we obtain whence A = — A, A = 0. 19, Theorem. — If each element of any column (or row') of a determinant is the stivi of two or more quantities, the determinant can he expressed as the sum of two or more determinants; tlius, (a/ + &/ + c/ + -) a/'...a/"> (a2' + V + «2' + -) a2"-"a2<'" (a„'+&„'+c„'+...) a„".-a„' (n) a/a," •••a,' 02' a^" •■•a2<"* (») + 6/ai"---ai<"' In I' ... riS") bjaj'. • a,' (n) + Ci'a/'.-.ai'"' c^'a," ■■■a./"^ (n) + • In the notation explained in Article 14, the given determinant may be written S± [(a,- + b,,' + c,; +...) a/' a,"' ... a/"']. This may be resolved into the following: 2± (a;a/'a,'"-a/">)+2± (&,V'«;"-«/'") + 2 ±(c,'a/V'-" «/'") + ■•■ 34 THEORY OF DETERMINANTS. [chap. III. These terms represent, in the notation of Article 14, the determinants in the second member of the above equation, respectively. Hence the theorem. 20. Theorem. — Multiplying each element of a column (or roiv) of a determinant by a given factor multiplies the determinant hy that factor ; thus. ma' o„ a, CO : m (n) [ a J a.'' • ■ ■ a,*"* a„ a„ a. (») Since each term of the expansion contains one, and only one, element from the column in question, if each element of this column be multiplied by a given factor, each term of the expansion will be so multiplied; that is, the determinant will be multi- plied by the given factor, which is the theorem. It follows that If each element of a column (or row') of a deter- minant is zero, the determinant vanishes. 21. Theorem. — If the elements of any row (or column) of a determinant have a common ratio to the corresponding elements of any other row (or column), the determinant is equal to zero; thus, ART. 22.] PKOPEKTIES OF DETEBMINANTS. 35 ai" maj ma^' •■■ ma.}"'> = 0. The common ratio (m) may be written outside the array as a factor of the whole determinant (Art. 20). The determinant then vanishes, because it has two identical rows (Art. 18). 22. Theorem. — If each element of a column (or row') of a determinant he multiplied by a given fac- tor, and the product added to the corresponding ele- ment of any other column (or row), the value of the determinant will not be changed; thus, a/ a/' fli"'...a/»' a,' a," o,"'...ai») a,' «+ma/") a/" •■• a/"' 02' {a," -{-ma,'") a^'" - o^'"' aj a.„" a,/"...a„<") aj («,/' + ma,.'") «„'"•••«,<"' The second member of this equation may be written (Art. 19), a/ a," . a,<") + «/ ma,'" a/" . . ai^^^ a,' a," • a/"' a,' ma,'" 02'" ■ ■ at^ «„' «„" • a,.'"' aJ ma,;" <" ■ ••«„•"' 36 THEORY OF DETERMINANTS. [chap. hi. The second of these determinants vanishes by the theorem demonstrated in the preceding article, and the given equation thus reduces to an identity. Hence the theorem. It may likewise be shown that we may combine, in the same manner as above, any number of columns (or rows) without changing the value of the deter- minant. Care must be taken, however, not to modify in any way the elements to which we add multiples of the corresponding elements from other columns (or rows). The above theorem in its general form is illustrated by the following equation : a/ ftj" «2"' 02" •••«2'"* a' a'' o'" a.}"- (») {a,' + laa" + ma,'"+-) a," a/' r,i") («„' + la„" + viaJ"-\ ) a J' a,,'" aj" ■ ■ ■ a„<"' 23. Theorem. — If a determinant is a rational integral function of a, and also a rational integral function of b, such that, if b is substituted for a, the determinant vanishes, then is (a — b) a factor of the determinant. For example, the determinant ART. 23.] PKOPEllTIES OP DETERMINANTS. 37 a— 7)1 na a' b —m nh V^ p q r contains the factor (a — 5). The expansion of the determinant may be written in the form A = Ao + Aia + A2a= + X3a'+"-, .... (1) in which Aq, Xj, Aj, ••■ are independent of a. Since upon substituting h for a the determinant vanishes, we have = X, + Xfi + \jy' + \,V + :. .... (2) Subtracting (2) from (1) gives A = Xi(a-6) + X,{a? - b') + \s{a' - ¥) + ■■■, which is divisible by (a — i^). Hence the theorem. The above theorem is often useful in factoring determinants without previously expanding them. Thus, the determinant As 1 a a' 1 b b' 1 c c' vanishes when a = b, when ffl = c, and when b = c, and therefore contains the factors (^a — b^, (^ci — c), and (h — c). Since the product of these three factors is of the same degree as A, these are readily seen to be the only factors. Hence A = (& — c) (a — c) (a — &) . 38 THEORY OF DETERMINANTS. [chap. III. EXAMPLES. 1. Tell the signs of the terms ajjfi^d^ei, OseiCidA, h^aie^Cids, h^c^Qidsei, afi^c^id^, hjii^^Cefii, e^dfija^fXi) and e^a^d^jj^Ci of the determinant | afi-^c^dies |. 2. Expand the determinant a e i m b f j n c g k o d h I p Prove the following equalities without expansion : 3. 5. 6. a, 6i ( 1 = - - «] ^■i a2 a2 bi Cj I 1 % &2 ttj &3 Cs c 1 "3 C2 Wlfli &i Ci : m «! 61 »! ma^ 62 C2 C3 h ffl3 maa 63 C3 C2 6, a. 2 12 + 3 2 6 = 20 1 2 3 3 1 1 6 3 9 3 1 15 1 1 10 3 7 5 1 6 1-7 = 0. 5 -10 5 4 3 - -7 ABT. 23] PROPERTIES OF DETERMINANTS. 39 -a + b + c a —b ':=• c a a — b + c b — c a b a + b — c c —a ' \b c am — ah)i \ = 0. - 1 bn 1 -m 9. Ee'HTite the determinant in Ex. 2 so that tlie element n shall occupy the leading position. Find the values of x which satisfy the following equations : 10. ja a x =0. 11. lo-l\r 11 10 = 0. Hi m m i 11 — o.r li 16 J6 .V b\ 12. I 1 sin a- =1. cos X sin x cos J- 1 I 13. Prove that 14 13 Ans. X = -• 4 1 J- -a y ~b = 1 X y = la- y 1 .r, — a y\ — t> ; 1 Xi .Vi Xy-x yi — y 1 .r. - a y,-b 1 J-,, y^ a-5 — J- ]/;-y 14. If 1 X y < = 0. 1 ^i Vi prove that -^- {X — Xj) . 40 THEORY OF DETERMINANTS. [chap. in. Eesolve the following determinants into factors: 16. 17. a a' b b" c c' be a' ac b" 1 ab c^ 16. 18. a a' be b b' ac c c' ab 1 a a^ a= 1 b b" ¥ 1 c c" c' 1 d (P cP 19. 1 a a^ a* 1 b b' b* 1 c c' c* 1 d d' d* Ans. {a+ b + c + d){a — b) (a — c)(a — d){b — c) (6_cZ)(c-d). 20. a h +c a' b a + c b' c a + b c- 21. a b c c a b b c a 22. Resolve the determinant A = 1 1 1 1 c' 6= 1 c' a' 1 b' a= into linear factors. ART. 23.] PROPEKTIES OF DETERMIKANTS. 41 Solution. — Let us designate by i* the ^th row and by LW the sth column of the given determinant. Then, dividing ij by he, is by ac, and i, by ah, we have A = a^'^e'^ 111 1 c b be b c 1 c a ac a c 1 6 a ah a b (1) Multiplying L' by abc, L" by a, L'" by 6, and Z'" by c, this becomes A = a 6 c a c 6 b c a c 6 a (2) This may now be put in the form a + b + c a b c a + b + c c 6 a + b + c c a a + b + c b a by taking for the first column L' + L" + L"' + i" (Art. 22), which shows that the determinant contains the factor (a + 6 + c) . In the same manner, by taking for the first column the sums L' + L"-L'"-L^,L'-L" + L"'-L'^, and L'-L"-L'i' + L'^, 42 THEORY OF DETERMINANTS. [chap. hi. it may be shown that the determinant (2) is divisible by (- a + 6 + c), (n - 6 + c), and (a + 6 — c), respectively. The product of the four factors thus found being, like (2) , of the fourth degree in a, b, and c, we infer that (2) is of the form A = X(a + & + c)(-a + 6 + c)(a - & + c)(a + 6 - c), (3) in which X represents a factor independent of a, h, and c. This factor may be found by comparing any term of the expansion of (2) with the corresponding term of the expansion of (3), say the term containing a*. The term a' in the expansion of (2) has the same sign as the term a.fifi^d^ of the determinant (1), Article 7, which is positive (Art. 12). The corresponding term in the expansion of (3) is — Xa'. Whence, X = — 1, and we have A =-(a + 6 + c)(- a + 6 + c)(rt- 6 + c)(a + 6-c). It appears that — A represents sixteen times the square of the area of the triangle having the sides a, b, and c. 23. Show that the determinant A = a b c d b a d c c d a b d c b a is equal to the product {a+b+c+d)(a + b — c—d)(a—b+c—d)(a—b—c+d). Note. — It may be shown that — A represents sixteen times the square of the area of the quadrilateral inscribed in a circle and having for sides — «, h, c, and d. ART. 23.] PROPERTIES OF DETERMINANTS. 43 24. By applying the theorems demonstrated in this chapter to the determinant tti 6, = 0, deduce the common theorems in proportion. 44 THEORY OF DETERMINANTS. [chap. ir. CHAPTER IV. DETERMINANT MINORS. To expand or evaluate determinants of higher than the second or third order, they are resolved into determinants of lower orders. The method by which this is accomplished is explained in the arti- cles immediately following. 24. In order to find the terms containing the ele- ment a/ of the determinant of the wth order, A = < a,'"' rJ") -I," <"-a,'"' x,/ c<„" a„"'...a„"" (1) let each element of the first row except a/ be zero ; the determinant thus becomes CTl' a,' a., a- a„ (11) (2) Each tenn of the expansion of this array contains as a factor one, and only one, element from the first row (Art. 12), and the only terms which do not ART. 24.] DETERMINANT MINORS. 45 reduce to zero are those formed by taking a/ as the factor from the first row. Hence, the expansion of (2) contains the terms of the expansion of A which involve a/, and it contains no other terms. Also, those terms of the expansion of (2) which do not vanish, can contain no other element than a/ from the first column. Each term is, therefore, formed by multiplying by a/ some one of the (n — 1)! products of (n— 1) elements formed by taking one element from each column and row of Ct'2 Cvg h" «3'"-a3'"' (») Any one of these products without the subscripts may be written a" a'".--a<">, and the different products may be formed by distrib- uting the (n— 1) subscripts 2, 3,---n in every one of the (w— 1) ! possible ways. The sign of the term foimed by multiplying any one of these products by a/ is determined by the class of the permutation of the subscripts 1, 2, 3, ■■■n, 46 THEORY OP DETERMINANTS. [chap. iv. and, since the subscript 1 remains in the first posi- tion, this permutation is of the same class as that of the subscripts 2, 3, ••• n. It follows, therefore, that the algebraic sum of all the pi'oducts which are multiplied by a/ is the determinant a," a,'" ... a^w (3) Hence, «3" a3"'-a3<"' a,' . • = 0/ 0," ri '" ■■ 02'"' a,' ai' a/" . . «2"" < a,'" ■ ..aJ-> < <' a„"'. • «„'"' a„" <'■■ ■ a„<'" (4) The determinant (3) is called the co-factor of the element a/ in the determinant A (Eq. 1). It may- be formed from A by deleting the" row and column to which the element csi' belongs ; that is, the first row and the first column. 25. The co-factor of any element, a^'"', of a deter- minant may be found by bringing that element into the leading position (Art. 17), and then delet- ing the first row and the fii'st column. The co-factor thus obtained will be positive or negative AM. 25.] DETERMINANT MINORS. 47 according as (k + s) is even or odd. But the process of bringing a^''' into the leading position does not in any way change the relations of the elements in the remaining rows and columns ; hence To find the co-factor of any element, a,;*", of a determinant, delete the row and the column to ivhich the element belongs, and give the resulting deter- minant the positive sign when (k + s) is even, and the negative sign ivhen (k + s) is odd. The co-factor of any element, a <"', of a determi- nant is represented by the symbol The sign of this co-factor is (-iy+', but the expression A^''' is generally considered as including the sign within itself, and is accordingly written as positive. The co-factors of the various elements of the determinant «/ ai" a, 0.2' a^" a. ttj' as" a. are as follows : 48 THEORY OF DETEKMINANTS. [CHAP. IV A,'= 02" 0,2" I -^1 — aj' 02'" A 'II — ttj' ■ a^" 03" <' tta' «3"' as' tta" A'=- a/' a/" , -a.2 — a/ a/" A III ) -"2 — — fli' a/' a^" %'" %' «3"' fs' tts" A,'= a/' Cl/" 5 -"3 — — a/ a/" A III > -^3 — a/ a/' 02" a^'" a^ a,'" 02' a^" 26. The result obtained by deleting the row and column to which any element of a determinant belongs is called the minor of the determinant with respect to that element ; thus the minor with respect to the element a^*'' is obtained by deleting the A;th row and the sth column. The co-factor of a^''' in the same determinant is (Art. 25) equal to this minor multiplied by the sign-factor (— 1)*"^'. The minor of the determinant A with respect to the element a^''' is represented by the symbol A". (* Accordingly, the co-factor A^^''' of the same element may be written (-i)'+-a;;. 27. The algebraic sum of the (w — 1) ! terms of A which contain the element aj'^ is a'^U'"'. ART. 27.] DETERMINANT MINORS. 49 The algebraic sums of all the terms which con- tain the successive elements a', a'\ a '" (n) '* ) ■•'* J "t ) ) "■* of the kih row, are respectively aJA,', a," A,", a •"A,"', -, a,<»>A"", n in number. Each one of these sums is com- posed of (ti — 1) ! of the terms of the determinant A, and no one of these terms is found in any other sum. There are then in all of them « X (n — 1) !, or w! terms, no two alike. They are the n I terms of the expansion of A. Hence A = a; A' + A"'>. (1) In a similar manner it may bfe shown that A = ai"Ui<" + a2<''^2"' + OgC' Js*"' + - + a„WA<''- (2) Bj/ means of FormulcB (1) and (2) we may express any determinant in terms of determinants of one lower order ; thus, applying Formula (1) to the determinant ai' < <' , (3) aJ a," a,'" 2s' a," a,'" gives, making k=l, 50 THEORY OF DETEEMINANTS. [chap. iv. A = tti' a," a,'" -a/' a,' a,'" + <' a,' «;' a," aj" aJ as'" C63' Cts" making /fc = 2, A = — tta' Ch" a,'" + a2" < a/" - a^'" ai' a/' a," %'" as' a^" as' as" making k = S, A = a,' a/' a/" -a3" a/ ar + as'" a/ ai" C6," a^'" a^ a,'" a,' 02" Formula (2) may be applied in a similar manner. Since, in Formulae (1) and (2), the co-factors are themselves determinants, they may be resolved into determinants of still lower order in the same manner. By this process any determinant may be expressed in terms of determinants of the third or second order, convenient rules for the expansion of which have already been given. 28. In the determinant A of the preceding article, if the row a*' a," ••■ a,("' be identical with the row 7 (») ART. 28.] DETEKMINANT MINORS. 61 then, instead of writing the expansion as in Equation (1), it may be written The above expression is, then, a form for the expansion of a determinant in which the Ath and yfcth rows are identical. Hence (Art. 18), when h and k are different subscripts, a,'A' + <'A" + <'A"' +•••+«/" W" = o. . . (1) Similarly, when p and s are different superscripts, a^p^A^''' + a^'"' A'"^ + ci3<^' A"' H h a„'^' A*'' = . . (2) Thus, in reference to the determinant (3) of the preceding article. ^^2 A + 0,% A + a J 0,. (") all the terms vanish except those which contain the element a/ from the first column (Art. 24). It follows that we may replace the remaining elements of this column, «2', by any quantities whatever, as B, T, without changing the value of the determinant; thus, 0/ •••0 Cto Cto Cttt do do Oto (n) (n) oj aj' aJ" < ... Q ai' a;" ••• a,("' R tta" aj'" ... ttjC" r a„" a„"' ••• «„'"> ART. 29.] DETERMINANT MINOES. = a/ a,y a,'" ■■■ a/"' as" ^3'" J"> x„" a„"' ... aj"^ 61 (1) If the elements aj'", a2''5 ••• ^2'"* f^lso become zero, we may reduce the determinant to one of the order (w — 2) ; thus, a/ 0-0 a.J a," ••• a„' a„" a.„"' ... a„w = a,' a^" ... as" «s"' - «s'"' aj' aj" ■■■ aj^^ a,' a," a,'" - ag'"' • .(2) a'" ... a ("> In this case the elements ^■3 , ^3 , Ct„ , rt„ , may be replaced by any quantities whatever, as Q, R, L, T, N. 62 THEORY OF DETERMINANTS. [chap. IV. In a similar manner it may be shown that if all the elements on one side of the principal diagonal are zero, the elements on the other side of the same diagonal may be replaced by any quantities whatever, and that the determinant is equal to its principal term; thus, a,' • ■ = o/ . ■ • aV a/' • ■ Q a^-'O • •• a,' a," as'" ■ • R L as'" • • • a„' a„" a„"' • •• «„'"* T N- H ■ •• a, = a/ a," a,'" ■■■ a„<'" .... (3) 30. By the preceding article we have m a/ ••• a/"' a ' •■• o„ m ••• '0 F aj' ••■ a/"' H a ' ... a (»> • (1) the capitals still representing arbitrary quantities. This furnishes another method of multiplying a determinant by a given factor. Letting m = I, we have a method of raising the order of a given determinant without changing its value. ART. 30.] DETEKMINANT MINORS. 63 That is A determinant of the nth order may he expressed as a determinant of the order (n + 1) by bordering it above by a row (or to the left by a column) of zeros, to the left by a column (or above by a row) of elements chosen arbitrarily, and by writing 1 at the intersection of the row and column thus added. By continuing this process, any determinant may be expressed as a determinant of any higher order ; thus, < ■•• a/"' o ' ••• a. (n) 1 . ■ ■ = F a/ • .. a/"' Ha: . •• an'"' 1 ••• K 1 ... L F Oi' ••• ai^"'' N H a„ (n) 10 ■ ■ P 1 .. Q iTl .. B L F a,' .•• ajW T N H a,; ■• «„<"' = etc. (2) EXAMPLES. 1. If, in any determinant A, a column of zeros be introduced between the sth and the (s + l)th columns and a row of arbitrary elements between the feth and 64 THEORY OF DETERMINANTS. [chap. iv. the (fc + l)tli rows, 1 being written at the intersection of the column and row thus added, show that the resulting determinant will be equal to (—1)'+' A. 2. Show that «/ «/' ■•• = «/ «/' X «,' .. a/"' «/ < ••• a^' «2" . a/ ••• a<^n) a,: ... «„"•> a„' ■■• an'"' 3. Show that • •• ai<"> = ... a/' a„_,"' - «„_/"' a„_ 2"'-" IF a„_." a„_i"' ••• a,.-i'"' a„_i"i ... iV" «„' «„" a„"' ... a„<") «„' Q R ... T = (-1 )?(" -"«„ 'an- /'«„ III n ( -2 • • • «1 (») 31. 7f k rows and k columns of a determinant of the nth order be deleted, the determinant of the order (n — k) formed of the (n — k)^ remaining elements* is called a kiA minor of the given determinant. * Deleting k rows removes kn elements ; then deleting k col- umns removes k(ii — k) more, after which there remain n^ — kn — k(n — k) = (n — ky ART. 32.] DETERMINANT MINORS. 65 If the deleted rows are, in order, the Ath, I'th, yth, ..., and the deleted columns the pth, qth, rth, ..., the resulting minor is represented by the symbol A (r. J. >■.■•■ '^ (», i. j. ■■■' in which A represents the original determinant. 32. Any kth minor of a determinant, and the determinant formed of the k^ elements at the inter- sections of the rows and columns deleted in forming this minor, are called, with respect to each other, complementary minors of the given determinant; thus, in the determinant A =\ ai' (h (h a" «; and ^(1.3,5 — at' tti" '^(2,4 - a/' a,'" «r a," a,'" (h^ a," a,'" as' are complementary minors. It may be remarked that any element of a deter- minant, and the minor of the determinant with respect to that element, constitute a pair of com- plementary minors. 66 THEOKY OF DETERMINANTS. [chap. iv. 33. Theorem. — The product of any two comple- mentary minors of a determinant is composed of terms which, signs being disregarded, are also found in the development of the given determinant. Let the given determinant, whicli we shall sup- pose to be of the wth order, be represented by A ; also, let /ij and /j,„_^ represent any two complemen- tary minors of A, these being of the orders k and (w — k') respectively. Transpose those rows and columns of A which contain the elements of /j.^ so that they shall be in order the first k rows and columns of a new determinant A'. If the number of necessary interchanges of rows be represented by u, and of columns by v, then will (Art. 16) A=:(-1)"+"A' (1) The determinant in its present form may be written A'= a/ (2) in which the determinant ART. 33.] DETERMINANT MINOES. 67 «i' x/« , (« occupying the upper left-hand corner of the array, is the minor fi^ referred to above, while the determinant «.+i<'+^' Hic+i) (n) (*+l) (n) occupying the lower right-hand corner, is the minor If the product be developed, each term of the development will be the product of a term in the development of fj,^ and a term in the development of fi^_„. But each term of the development of yUj contains an element from each of the first k rows and one from each of the first k columns of A', and each term of the develop- ment of /ii„_j contains one element from each of the last (n — k') rows and columns of the same determi- nant. Thus each term of the developed product, /"iMn-H contains one element from each of the n rows and columns of A', and is therefore a term in the development of A'. Hence (Eq. 1), signs being disregarded, /Xi,fj,„_t is composed of terms of the expansion of A, which was to be proven. 68 THEORY OF DETERMINANTS. [chap. iv. 34. It follows from the above that, signs being duly regarded, the product (-l)»+>,|ot„_, is composed of terms of the development of A. Now, as may readily be seen, the only factors by which fj,^ is multiplied in the development of A are the terms of the development of yu„_j and the sign factor, ( — 1)"+". Hence, in the determinant A, the minor fi^ has the co-factor ( — 1)"+" fj,^ ^. We may therefore state the following Theorem. — The co-factor of any minor of a deter- minant is the complementary minor multiplied by the sign factor ( — 1)""^', in which u is the number of interchanges of roivs, and v the number of interchanges of columns, necessary to bring the principal diagonals of the tivo minors into coincidence. 35. If /lij be the minor A (P. ?. r,... of the determinant A, the principal diagonals of /^j and its complementary minor /i„_t may be brought into coincidence by (p-l) + (g-2) + (r-3)+... interchanges of adjacent columns, and ART. 36.] DETERMINANT MINORS. 69 {h-l) + {i-2) + U-S) + - interchanges of adjacent rows. Hence the sign factor of the complementary minor A (P, «. <■<■■■ (_l)(p-l)+(«-2)+(r-3)+-+(»-l)+(i-2)+0--3)+-^ which is the same as of is 36. The co-factor of a given minor of a deter- minant is often expressed by prefixing the syllable CO- to the given minor ; thus, in the determinant A = a,' a," . the co-factor of the minor a,' < a,' at may be written co- a,' ar a; ar By the two preceding articles we have CO- at' at" = co-A, 73 in which 5/'' has the sign (-!)'+• . . . (7) Cauchy's method is useful in expanding determi- nants which have been bordered, a class of determi- nants quite common in analysis. We take, as an example, the bordered determinant A = 1 I X a. m n a/' a,'" 2 03' as" tts'" The expansion by Formula (6), or (7), is a/ ai" a/" -Ix a," a,'" + mx ttj' ttj'" —nx a^ a^" a; aj," a/" n " n '" tts' as'" as' as" a^ according to the elements of the first column and the first row. 76 THEORY OP DETERMINANTS. [chap. iv. 39.* Let it be required to find the differential of the determinant A = • M X^M SO,' a:,. (n) x,^'^ is By Formula (1) of Article 27, A = X.V + X,'%" + ■■■+ X,<"'a;,<"'. The differential of this with respect to the element V>A = AV"dxW; (1) that is, the differential of a determinant with respect to any element is the co-factor of that element multi- plied hy the differential of the element. The total differential taken with respect to the elements of the \.th row is the sum of the partial differentials with respect to the elements of the ^th row ; that is, d,„ A = X.'dxV + X^'dx^' + - + X,<"'da;,(">, or, rfwA: (2) * The reader who is not familiar with the differential calculus may omit this article. ART. 39.] DETERMINANT MINORS. 77 The total differential taken with respect to all the elements of the determinant is dA = d(i,A + d(2)A H h d(„)A, or, by the preceding equation, dA: a;,' X," ■■• a;,<") xj x„ (71) + a-,' Ki" •■• a;/"' cZko' dxJ' ••• da;,'"' («) + • + r/ a;/' i (n) , (n) da;,/c?a;'' ... da;'") .(3) In the same manner we may derive the formula dA = dx/ aV • .. a;,<") + a;/ da;," • ■ a;i<») dx,' X," . .. x,<"' a;,' da;2" • . x^C) dxj a:„' dxj' . • a;,/"> + + a;i' a;/' ■•. da;,'"' a;;;' a'a" ... da;2<"' xj xj' ... da;,.<"' (4) 78 THEORY OF DETERMINANTS. [chap. iv. 1. Given 2. Differentiate EXAMPLES. X y a2 &2 =0;find ^. dx sin X sm y cos X cos y Ans. bi — & 2 % — a2 3. Differentiate 1 X 1 y a; 1 y 1 1 y 1 a; y 1 a; 1 4. Differentiate 5. Differentiate yz xz xy yz 1 1 1 xz 1 1 1 xy 1 1 1 1 COS 2 COS 2/ cos 2 1 COS a; cosy COS a; I 6. Given A = a h 9 X h b f y 9 f c z X y z ART. 39.] show that DETERMINANT MINOKS. d^ 79 = -2{Ax + Hy+Gz), = -2{Hx + By+Fz), and dx dA dy ^ = -2{Gx+Fy + Cz), dz A, B, C, F, G, and H being co-factors of the elements of a h 9 h b f 9 f c 80 THEORY OF DETERMINANTS. [chap. v. CHAPTER V. APPLICATIONS OP DETERMINANTS TO ELEMENTARY ALGEBRA. Enough of the theory of determinants has now been developed to enable us to study, by its means, some of the fundamental properties of systems of equations. It is with this application of the theory that the present chapter is principally concerned. 40. Let it be required to solve the simultaneous linear equations a/a;' + a/'*" + ai"'x'" = fcj, a,'x' + a,"x" + a.J"x"' = k^, The above equations holding good, we may write, arbitrarily, the equation (tti'a;' + a/'x" + a,'"x"') a," a/" {ajx' + a^'x" + a,y'x"') a," a,'" (as'x' + a^'x" + a^"x"') a," a^'" II „ III ki tti" ai «;,o tto C(; III By Articles 19 and 20 the first member of this equation may be written ART. 40.] APPLICATIONS. 81 a/ a/' a,'" a^ a^" a. 0t3 Ct3 d^' + x" a/' cf/' a/" +a;"' a^" aj" a^'" ri " n " n '" ftj O3 Oj a/" a/' a/" a,'" %" a,'" % C's ttj each term of which, except the first, vanishes by Article 18. The preceding equation thus becomes < a," a,'" = fci a/' a/" a,' a," a./" fc^ a." 02'" as' a," n '" ^3 "'2 as" a/' ^3'" a/" a; = ftg «3" fts'" a/ a/' a,'" rta' «2" a"' as' a," a,'" , or The values of a;" and x'" may be found in the same manner. This method is leadily seen to be a general one for the solution of systems of linear equations. Let us now proceed, in exactly the same manner as above, to solve the following system of n simul- taneous linear equations, involving the n unknown quantities, x', x",x'", ■•■ a;'"* : 82 THEORY OF DETERMINANTS. [chap. v. tts'a;' + a^'x'< + ••• + a3<'-"a;"-^' + a3«a;<''> + a3<*+«a;('+« }- (1) + ...+a3""a;(") = A;3, a„'a;' + a,."a;"+... + a„'*-"x"-" +a/'x<"' +a„('+"a;<'+" Let a;*" be the unknown whose value is sought. The above equations holding good, we may write, arbitrarily, the equation a^a^'--a^^'-''\a.^x'-\ |-a2'''a;«H l-a2<"'^<"')a2"+"-"a2"" ai'a/'"-ai<'-"A;iai""'"---ai"" .... (2) a2'a2"---a2"-"A;2a2('+«...a2<'" a„'a''.-a,('-"fc,.a<'+»-a<"' '*'n ^"ii ^"» "'7i"'7t By Articles 19 and 20 the first member of this equation may be written ART. 40.] APPLICATIONS. 83 a/oi" ••• ai('-»ai' a/'+i' Uo'tto' + ■■■ + a^'a^" + •■■+»' (n) a <'-"«„' a ('+" a/") a, (») a, a2"-i'a2« a2<'+" ••• a^'"' a 'a ''-a ('-"a'"' a '^+i> - a <»> Each term of the above expression, with the excep- tion of the one containing x^'\ vanishes by Article 18. Hence Equation (2) becomes K<') (n) a2>' (n) 84 THEORY OF DETERMINANTS. [chap. v. a^'a," ■ •«!< -'% a/'+" • • ai<"> cu'a" • a2< ''% a2<'+" . • a.^"* ajaj' • ■ «„< -"fc,. a„"+" . JaJ' ■■■ a <'-"o,(') o J'+i' ••• a<'" . . . (3) An inspection of Equation (3) enables us to state the following Theorem. — (a) The common denominator of the fractions expressing the values of the unknowns in a system of n linear equations involving n unknown quan- tities is the determinant of the coefficients, (b) The numerator of the fraction expressing the value of any one of the unknowns, is a determinant which may he formed from the determinant of the coefficients hy sub- stituting for the column containing the coefficients of this unknown a column whose elements are the absolute terms of the equations, these being taken in the same order as the coefficients which they displace. In Articles 3 and 6 we have already met with special cases of the above theorem. ART. 40.] APPLICATIONS. 85 EXAMPLES. 1. Solve the simultaneous equations, — x + %j + z + iv = S, x — y + z + w = 6, x + y — z + w = 4:, and x-\-y-\-z — iv — 2. 2. Solve the simultaneous equations, a; + 2/= 4, y + z=: 8, 2 + w = 12, and w+ x= 8. 3. Solve the system of equations, x + 2y-z= 0, y + z — 2w= 0, — x + z-\-2w = 25, and 2a; — 52/ + 3w = — f. 4. Find the values of the unknowns in the system, x + y + z = 0, 3x + 2y + z = 0, and 2y + 3z = 0. 86 THKOUV OF DETKUMINANTS. [chap. v. 5. Solve the system, x + y — z = 0, x + 4:y — 3z=0, and ox — ]/ — z = 0. (See Art. 42. ) 6. Representing the expansion of the rational fraction g,, + a,x + a^ + a^v? + • • • \ + h-sc + h^ + h^x' + • • • in ascending powers of x, \i^ qfs-\-q-^x-\-q^x'^ -\- q^y? -\ , show that Of, h • ■ tto h h ■ ■ • % h h b, ■ • • a^ h h h ■ • • as K &„-! K-2- -h «„ Suggestion. — Clear of fractions and equate coefficients. 41. If the number of equations in a given sj'^stem be greater than the number of unknowns, it will not, in general, be possible to assign to the unknowns values which will simultaneously satisfy all the given equations. Whenever values may be assigned to the unknowns which will simultaneously satisfy all the equations, the system is said to be consistent. ART. 41.] APPLICATIONS. 87 The consistency of any system containing a redun- dance of equations must obviously depend upon some relation among the coefficients. We now pro- ceed to investigate this relation in the case of (n + l) linear equations involving n unknowns. In this case the equations may be written, ai'x' + a/V H \- ai'"'a;' (n)„,(n) :A;i a^'x' +a2"x" -\ h aa'"'^;'"' =h ajx' +a„"x" H 1- «„'"'«<"' =K a„+i'x> + a„^,"x" + ■■■ + a^^.Mx^"^ = k„,. 1- • • ■ (1) Since the above system is to be regarded as con- sistent, the values of the unknowns obtained by solving any n of the equations must satisfy the remaining equation. Solving the last n equations by the method ex- plained in Article 40, we obtain k2 k. aj ■■■ a.,(") k„ 0, . (2) which is the required relation among the coefficients; that is, the condition of consistency. ART. 42.] APPLICATIONS. 89 Hence, the condition of consistency for a system of linear equations involving a number of unknowns one less than the number of equations is that the deter- minant of the coefficients and absolute terms shall be equal to zero. When the equations are consistent, this determi- nant is called the eliminant or resultant of the sys- tem, because it is the result obtained by eliminating the unknowns from the given equations. 42. The case of a system of n homogeneous linear equations involving n unknowns is closely related to that just considered, but demands special con- sideiation. In this case the column of absolute terras (^'s) becomes a column of zeros, and the numerators of the fractions giving the values of the unknowns vanish. (See Arts. 40 and 20.) Hence a system of homogeneous linear equations will always be satisfied by giving to each unknown the value zero.* Example 4 after Article 40 furnishes a case in point. There are, however, systems of this kind in which the equations may be simultaneously satisfied by assigning to the unknowns values other than zero. These cases we shall now consider. * This is, of course, obvious on other grounds, and is true for homogeneous equations of any degree. 90 THEORY OF DETERMINANTS. [chap. v. Let «„'»' + aj'x" H H a„('''x('" = ) (1) be any system of n homogeneous linear equations involving the n unknowns x', x", ■■■ x^"\ in which the coefficients are so related that a/ a/' ••• a{ J aJ' ... aM = 0. (2) Applying the method of Article 40 to the above sys- tem, we can obtain the value of each unknown only in the form ^, which may have any value whatever. Though it is thus impossible to determine the absolute values of the unknowns in a system such as the above, it is possible to find the ratios of any (?i — 1) of the unknowns to the remaining one. For, dividing each of Equations (1) by a;'" and rep- resenting the ratios X' X" X' :(") :«)' 3.C.)' M. X'-" X'-' - by v', v", ■■• 1)'"', respectively, remembering that w'*' = 1, we obtain the system of equations ART. 42.] APPLICATIONS. 91 aj'v'-\ |-ai'*"^''y'''""+ai<'+%(''+''-| |-a/"'i;<">= —«/■'> ' «»''«' + ••■ +ai'""w'''"" + aJ'+'V'+» + - + a„("V"*=-a„« (3) This is a system of n non-homogeneous linear equations involving (n — 1) unknowns, v', ■■• v""", ^(•■+1), ... -yf"), and the condition expressed by Equa- tion (2) holding good, the system is consistent, and the values of the ratios v', •■■ v^'~^\ i)**"*"", ••• d<"' may generally * be obtained by solving any (n — 1) of Equations (3). Hence, Equations (3), and conse- quently Equations (1), vi^ill be satisfied by any values of x\ x'\ ■•■ a;'"' among which we have the ratios v', v", ■•• v'"', as determined by any (n — l) of Equations (3); that is, if Equations (1) are sat- isfied by the values Xo, x^", ■■• a;o'"\ they will also be satisfied by the values \xo, \xo", ••• Xa;o*"', X being any factor whatever. Since only (n — 1) of the n equations (3) are re- quired to determine v', v", ••• d'"', it follows that in any system of (n — 1) homogeneous linear equations involving n unknowns, the values of the ratios of these unknowns may be determined. If there are n equations in the system, the vanishing of the determinant of the coefficient, shows that * The only limitation is that there shall be at least one element in the determinant (2) whose co-factor does not vanish. 92 THEORY OF DETERMINANTS. [chap. v. the same values of the ratios «', v", •■■ d^ may be deduced from any (w — 1) of the equations j that is, that the equations are consistent. Hence, the condition of consistency for a system of n homogeneous linear equations involving n unknowns is that the determinant of the coefficients shall be equal to zero. When such a system is consistent, the determinant of the coefficients is called the eliminant or resultant of the system ; thus, the determinant (2) is the eliminant of the system (1). By way of illustration, we resume Example 5, after Article 40. We have 11-1 14-3 5 -1 -1 :0; that is, the determinant of the coefficients vanishes. This gives (Art. -40) ■^ 0' 0' results which the student has already obtained ; hence the equations fail to determine the values of the unknowns. The vanishing of the above determinant, however, shows the system to be consistent. ABT. 43.] APPLICATIONS. 93 To find the values of the ratios - and -, let the z z equations be divided by z. They thus become ^+ y=i > , 2 2 5^- y=i any two of which give H H '- -^- = = 1 = 2:3, and any quantities having these ratios will satisfy the given equations. 43. Having given the relation A= a/ ai" ••■ ai'"' = 0, " ... n (") then, by Articles 27 and 28, we may write aMt + a "A" +■■■+ a/"' A"" = aM.' + «;'A"+ - + "*'"M,'"' = = A aJA,'+ a„"A,"+ ••• + a„""A"'' = >, 94 THEORY OF DETEKMINANTS. [chap. y. in which the values of the ratios A""'' A^'^^^ ^<'' ' ^'••' ' ^ <"' no matter which subscript h may be, are the same as the values of the ratios „(■•-!) „,«+!) „(0 ' ,,(.■) ' „W' given by the equations a/a;' + «i"a;" H h ai<">a;<"' = o„'a;' + a,!'x" H h «,/»'»;<''' = (1) Therefore x\ x", ■■■ a;*"' are proportional to A,,', At", •■• At *■, and giving to k the values 1, 2, ••• w in succession, we have x' -.x'- : - :a;<"' : : A,' : A," : ••• : A^^"^ ^ : : A^' : A2" : ■■■ : ^2<»' An '■ A- : ^("> •(2) Hence, in any determinant which equals zero, the co-factors of the elements in any row (or column) are proportional to the cof actors of the corresponding elements in any other row (or column). ART. 44.] APPLICATIONS. 96 44. Of the proportions (2) in the preceding article let us consider the last, viz. : a;<") ■.-.AJ: AJ' : : A'»'- (1) The coefficients of the last of Equations (1) of the same article do not appear in the co-factors A„', A„", ••• AJ"\ and it follows that the proportions (1) give the ratios of the unknowns x', z", which satisfy the (w — 1) equations A") a/x' +ai"x" H |-ai<"'a;<"> =0 a„ 1^' + an-i"x" +■•■+ a„_i(">a;"" = . (2) expressed in terms of determinants formed from the coefficients of these equations by suppressing the column of coefficients of each unknown in turn; thus, if we place !)(■' = (-1) -IV'-" a/'-" ai<*+" • ■ a (n) «„-l «„-l «■„_! Mn-l (3) the solution of Equations (2) is x' : x" : — : «<"> :: D' : D" : ■■• D'"'. • (4) For example, the two equations !=0) a; + 4y — 3^ = I 5a;— y — z= 96 THEORY OF DETERMINANTS. [chap. v. give X : y : z ■.: 4 -3 :- 1 -3 : 1 4 -1-1 5-1 5-1 : : 1 : 2 : 3. 45. Substituting in Equations (2) of the preceding article the values of the ratios a;' X" ,(t-i) „.(i+i) „(0 ' a;"' given by the proportions (4) of the same article, we obtain the (w — 1) equations a/Z>' +a^'D" +-+ai<">i)("' =0, a^D' +a^"D" + • • • + a^*"' -D'"' =0, a„-i'-D' + a,.-i"X>" + - + a„_/">i>'"> = 0. J These (n — 1) relations are expressed by writing a,' rti" (I2 ^2 (n) ,.(») a„-i' a„ .(n) = 0. (2) This expression is called a rectangular array or a matrix.* * The rotation ( ai' oi" •■■ (ii<") ) = O2' (12" •■• 02*"' a,,' a„" ••■ a„(»' has of late years come into very general use. The discussion of the general theory of matrices is not ■within the scope of the present work. ART. 46.] APPLICATIONS. 97 In the rectangular array (2) the number of col- umns is one greater than the number of rows. In this case we have, by Equations (1), a." a. a," a. (.1) ^11-1 O^n-l ••• Bn -l--+(-i)'' V" .(") J") «„-/ CI I ,, III h' «l" (l.-l) (1,-1) \-l f'n-l «„-/""'' =0, . . (3) in which k may have any integral value from 1 to (n — 1) inclusive. Thus, in reference to the rec- tangular array = 0, a-i &i Cl t\ 0,2 h C2 6,2 as h Ci di we have «i I ^iC-'C^s I — &i I o-iC^A I +Ci t «i&2<^3 1 — c^i I aiVs I = 0, 0-2 I &lC2f^3 I —^2 I afii'^S I +C2 I fliMs I — <^2 I «l&aC3 | = 0, (h I h^id^ \—h\ aiC2d3 1 +C3 1 ai62C^3 1 — f's | ai&sCs 1=0. 46. Let us assume r homogeneous linear equations involving n unknowns, r being greater than n. They may be written as follows : 98 THEORY OF DETERMINANTS. [chap. v. a/a;' + a/'a;" H h a/"'a;<"' = 0, ^ a^x' + a^'x" H h a2'"'a:<"* = 0, , . . . (1) ajx' + aj'x"^ |-a„("'a;"" = 0, ' ajx' + aj'x" H 1- a/"'a;<"' = 0. _ If these equations are to be consistent, then, by Article 42, the determinant of each and every system of n equations taken from the above r equations must be equal to zero; that is, every determinant which may be formed from any n rows of the array a/ .. a/") a^ ■■ a2'»> a„' ■■ «„<"' a; .. a/"' must be equal to zero. The existence of this relation among the elements of this array is expressed by writing a.' •• a J ■■ as'"' a„<") = 0, (2) AKT. 46.] APPLICATIONS. 99 or, changing (arbitrarily) rows into columns and columns into rows, by a/ o.; a^'"' (") «;■ = 0. (3) The condition of consistency for a system of r linear* equations involving n unknowns, r being greater than n, is thus concisely expressed in the form of a rectangular array. As an example, take the equations 2x + y — 2 = 0, x-2y+ 2 = 0, x + 3y-2z = 0, 4a; — 3?/+ z = 0. These are consistent because 2 1 -1 = 0, 2 1 1 -2 1 1 -2 1 3 -2 4 -3 2 1 -1 = 0, 1 -2 1 3 -2 1 3 4 -3 1 4 —3 -1 1 1 1 -2 1 =0, •(4) * If the equations are not honlogeneous, then ai*"', 02'"', ••• a„<"', •■■ Or'"' may represent the absolute terms, and the same conditions will still apply. (See Art. 41.) 100 THEORY OF DETERMINANTS. [chap. v. These four relations are all expressed by writing 2 1 1 4 =0. 1-2 3-3 -1 1-2 1 By Formula (3) of Article 45 we should have 114- -2 3 -3 1-2 1 2 1 -1 2 1 3 -2 1 1 -2 -1 1 + 2 1 1 -2 -1 1 :0, with two other equations of a similar character, all of which may at once be seen to hold good on account of Equations (4). EXAMPLES. 1. Test the consistency of the equations, x+ y + 2z = ^, x+ y— 2 = 0, 2a;- y+ 2=3, and x — 2>y + 2z = l. 2. If the equations X— y — 2z = 0, x-2y+ 2 = 0, and 2x — 3y— z = are consistent, find the ratios x:y:z. AKT. 46.] APPLICATIONS. 101 3. Pind the ratios of the unknowns in the equations — 4.1;+ y + z = 0, and x — 2y + z = 0. ■ 4. Find the ratios of the unknowns in the equations 2x+ y -2z =0, y + 4:Z — iiv = 0, and x—5y+ z+2w = 0. 5. From the equations ax + yy + I3z _ yx + by + az _ fix + ay + cz I VI n deduce the relation x y % y /3 I a 13 I b a m y a m a c n P c n a y I y b m P a n 6. In the equations a^'x' ^ y. a.i<''-^'a;<»-" + a/"' = 0, «n-/^' + - + a„_/"-»a;("-"+ «„_,<"'= 0, show that x' : x" : ••• : a;'"-" : 1 : : D' : D" : ••• : Z><"-" : J9'"'. 7. Solve the equations 2x+ y + 3z = 19, x- y + 2z= 7, and 3x + 2y- z= 8, by the formula given in the last example. 102 THEORY OF DETERMINANTS. [chap. v. 8. If any one of the equations of a system of n non-homogeneous linear equations (or of n — 1 homo- geneous linear equations) involving n unknowns is a consequence of the others, show that the equations fail to determine the values of the unknowns (or of their ratios). 9. Solve the equations x+3y — 2z = l, 5x + 5y — 2z = 9, and 3x— y + 2z = 7. 47. The principles of the preceding articles may be applied to the solution of the problem of elimi- nating a single unknown from two consistent equa- tions of any degree. We shall explain two methods of dealing with this important problem. The simplest is Sylvester's dialytic method, which is as follows : Let the two consistent equations from which it is required to eliminate the unknown be ttaX^ + ttiX' + a^x -1- oo = 0, I bix' + b^x + 6u = 0. J Multiplying the first of these equations by x, and the second by x and a? successively, we have the five consistent equations ART. 47.] APPLICATIONS. 103 a^o^ + a^y? + a^x + Oq = 0, a-^x* + a.p? + ajx-^ + afpo = 0, 62*;^ + &!« + &o = 0, }- ftja^ + hyX^ + 6(,a; = 0, (2) These five Equations involve tlie four unknowns 2:*, x^, a;^, and x. Then, by Article 41, their elimi- nant is Oj a2 Ui Oo = 0. . . . (3) a-i 02 cti tto 62 &i 60 b, b, h 62 61 &o Equations (1) being consistent. Equation (3) is their eliminant; or, it being uncertain whether or not Equations (1) are consistent, Equation (3) is the condition which must be fulfilled in order that they may be so. In general, let the two equations be a„a;'"+a„_ia;'"-'+- M"+^.-ia''""' + ---+&2a!'+M+&o=0 (4) Multiplying the first equation by x, x^, •••, and x"~^ successively, and the second by x, x^, ••■, and a;'"~\ we have (m+w) equations similar in form to Equations (2). They may be written as follows : 104 THEORY OP DETERMINANTS. [chap. t. (i„a;'»+"-i + - — |-a,a;"+aoa;"-^ =0 finK" + 6n-iK"-i H 1- 6iX + 6o = 6„a;'»+"-i+"-+ fciaj^+io^"" ' :0 (5) These (m + w) equations involve the (m + w — 1) unknowns, 0;"+""', •••, a;^ and a;. Then, by Article 41, their eliminant is •■•0 a„ a„_i--- •••«„ a„-i «„-2--- •■•0 ."0 6„ ■•■ &2 &i \ a., a. n-2 «i Oo «! Oq &„ &„_!••• 62 &1 &0 &„ Z'„_l &„_2-" Z*! h =0...(6) This determinant is of the order (m + n). It is of the nth degree in the coefficients of the equation of the mth degree, and of the mth degree in the coefficients of the equation of the nth. The same metliod will apply when the two equa- tions are homogeneous in two variables and it is required to eliminate either one or both variables. ART. 47.] APPLICATIONS. 105 As an example, let us eliminate x and y from the two equations 2 a^2/ — xif = 0, Sx'y +8xy'-5y* = 0. Dividing the first equation by y^ and the second by y*, they become r y :0, + 8--5 = 0. y X X Multiplying the first by - and by -^, and the second by -5 we obtain, as from Equations (1), the eliminant = 0; 2 -1 2 -1 2 -1 8 8 -5 8 8 -5 a condition which, being fulfilled, shows that the given equations are consistent. Sylvester's dialytic method may sometimes be applied in eliminating the unknowns from more than two equations. The following example, due to Professor Cayle)s will illustrate. 106 THEORY OF DETERMINANTS. [chap. v. Let tlie given equations be x+ y + z = 0, 3?= a, and 2^ = c. Multiplying the first equation by xyz, by x, by y, and by z, we obtain, by the last three equations, ayz + bxz + cxy = 0, a + xz + xy = 0, 6+2/3+ a;2/ = 0, c + !/2 + .r« = 0, from which yz, xz, and xy may be eliminated as sep- arate unknowns, giving a b c a 1 1 b 1 1 c 1 1 = 0. Or, multiplying the first equation by yz, xz^ and xy in succession, we may obtain the eliminant in the form = 0. 1 1 1 1 c b 1 c a 1 h a ART. 48.] APPLICATIONS. 107 The eliminant of the equations x + y + z = 0, a? = a, f=b, and z^=c, . may be found in a similar manner. 48. We shall now consider Euler's method of elimination. Let the given equations be F{x) = a^T? + a^ + ttjX + Oq = 0, /(x) = 620^ + biX + &„ = 0. The consistency of these equations requires that they have a common root, which we shall represent by r. Then Fix) „ , x—r -^M= Aa;+;8o, (2) x—r in which «2, «i, «o' ^v ^"'^ '^o ^^^ undetermined. Equations (2) give (a^x' + ttsO^ + aiX + ao) (/Sja; + ;8o) 108 THEORY OF DETERMINANTS. [chap. v. Equating the coefficients of the like powers of x in the two members of this equation, we have &2«2 — 03^1 = 0, ftlKj + ^2«1 — Ct2jSl — «s'?0 = 0) 6o«2 + ^i«i + ^2'^ — «ij8i — 02/^0 = 0) 6o«i + 6i«o — floSi — ai^o = 0, h„tt.fi — ao/?o=0. (3) Eliminating a^, «j, Kq, ySj, and /3o from the above system gives 62 aj = 0, (4) &0 &i 62 ffl ^^2 60 ^1 «0 «I &o Oo which is the eliminant of Equations (1). This determinant is the same as the determinant (3) of Article 47. In general, let the given equations be ^(a;) = a,„a;"H h aja;^ + ajo; + ao = 0, f{x) = 6„a;" + ... Jrhx' + biX+bo =0. Tiien, r being a common root of these equations, we have ART. 48.] APPLICATIONS. 109 = (x^-iaj^-'H 1- aia;+«o = *(«), li^-r, 1 x—0- x—r in which a„_i, ..., «i, «(,, /3„_i, ..., ySj, and /S,, are (m + n) undetermined quantities. These equations give F{x) . {x) =f{x) . ^(x), each member of which is of the degree (m + w — 1) in X, and by equating coefficients we may obtain (m + «) equations which are linear and homogeneous with respect to the (jm + n) undetermined quantities «„_!, ... ag, ;S„_i, ... /Sq. The determinant of the coefficients of these (m + n) equations is the required eliminant. This determinant is the same as the determinant (6) of Article 47 except that columns of the one appear as rows in the other, and vice versa. By means of Euler's method we may readily find the conditions which must be fulfilled in order that two equations may have two or more roots in com- mon. For example, let the equations be F{x) = a^x^ + rtoX- + a,a; + a,, = 0, f{x) = hx' + b.^' + b,x + bo=0, the common roots being j\ and r^. 110 THEORY OF DETERMINANTS. [chap. v. Then, as before, : aiX + «o, = l^ix + /3o. {x—i\){x-ri,) These equations give (ttsx' + a.{s? + a^x + ttu) (/Si* + ^o) = (h^o? + h^i? + h^z + 60) («ia; + «o) ■ Equating the coefficients of the like powers of x in the two members of this equation, we obtain fts/?! — &3«1 =0) «2,8l + Qs^o — &2«1 — ^3«0 =0, «l/3l + 0«i% — &1«1 — ^2«0 =0, «oA + «l/3o — &(l«l — &1«0 =0, «o5o— 6oao=0. These five equations, being homogeneous and linear with respect to the four undetermined quanti- ties /Sj, ;Sq, «!, and ag, the conditions of their consist- ency, that is, the conditions which must be fulfilled in order that the given equations may have two roots in common, are (Art. 46) expressed by the rectangular array flo «1 ^2 "s a,) ai a, (% hn hi ^2 &3 ia ^/i &2 h. ART. 48.] APPLICATIONS. Ill The student can readily generalize this process for himself. EXAMPLES. 1. Test the consistency of the equations and a^-2x -3 =0, by Sylvester's method, and also by that of Euler. 2. Test the consistency of the equations 6x^-3x-y- xy'--12if = 0, and Aa^ —8xy + 3?/^ = 0. 3. Eliminate a; from the equations in the last ex- ample, thus obtaining an equation in y only. 4. Eliminate x and y from the equations aacFz + bxy + c = 0, dyz + ea; = 0, and fxz + gx+ h = 0, thus obtaining a single equation in z. Ans. az b c e dz (fi + 9) h ifz+9) h ifi+g) h = 0. 112 THEORY OF DETERMINANTS. 5. Free from radicals the equation [chap. v. VoiS; + ao + ■\/hyX + &„ + Co = 0. Solution. — Let OiX + flo = «/■=, and &ja; + 6(, = z^ The given equation thus becomes 2/ + 2 + c„ = 0. . . Eliminating z from (2) and (3) gives 1 - (6,a; + 6„) = 0, 1 (2/ + Co) 1 Oy + Co) r !/2 + 2c„!/-(6iX + 6o-c„2)=0. . . . Eliminating ?/ from (1) and (4) gives 1 - (o,x + flo) = 0, 10 — (fliX + rtj) 1 2 Co - (fiiK + &o - Co^) 1 2 Co -(fiiX+fto-Co'^) vfhich is an equation in x and free from radicals. 6. Eliminate the unknowns from the equations x^ = a, (1) (2) (3) (4) ^KT. 48.] APPLICATIONS. 113 Ans. One form for the eliminant is 100011000=0. 010 101000 001110000 Oc &000100 cOaOOOOlO daOOOOOOl 000 a. 00011 OOOO&OIOI OOOOOcllO 7. Find the condition that all the roots of the equa- tion x' + SHx+G^O shall be real. Solution. — The three roots may he represented by u, ^ + Vt^ and ^ — v^ These will all be real when 7^>0, (1) and the last two will be imaginary when 7''<0 (2) From the well-known relations between the roots and coeffi- cients of the rational integral function, we have 2 0/3 + ^2 _ .j,2 _ 3 ^^ 0/32 _ a.y2 = _ Q_ (3) 114 THEORY OF DETERMINANTS. [chap. v. Eliminating o and /3 from Equations (3) gives 3 (72 + 3 if) 3 (72 + 3Jr) 3 (7= + 3^) 2 -272 2 -272 -G -G = 0. This readily reduces to 27 ©2 + 4 £-3 4 (4 72 + 9i/)2' which, compared with (1), shows that the roots are all real when (?2 + 4 £-3 < 0, the required condition. When (?2 + 4fi'3>0, the two conjugate roots are imaginary. The function ©2 + 4 H^ is called the discriminant of the cubic x^ + ZHx+ G = 0. 8. Find the condition to be fulfilled in order that two of the roots of the equation may be equal. Solution. — Let a, o, and p be the three roots, and we readily obtain 2a + /3 = -3a, a2 + 2 afl = 3 6, o2^ = - c. From the first of these equations ^=-(3a + 2a), AKT. 48.] APPLICATIONS. which reduces the second and third to a2 + 2 aa + 6 = 0, 11-5 respectively. 2a8 + 3aa2 c = 0. Eliminating o from these last equations by Sylvester's method gives 00 12a 6=0, 1 2a & 1 2o 6 2 3(1 -c 2 3a -c or, 4a^c-3a'b^-6abc + ib^ -^ c^ = 0, . . . (1) the required condition. Or, since a double root is common to the given equation and its first derivative, we may apply Euler's method of elimination to the two equations x^ + Z ax^ -\- Zbx + c = 0, x- a;2 + 3 6x + c = 0, ] X- + 2 aa; + 6 = 0. J (2) We should thus obtain the same condition as before. 9. rind the double root of the equation when the condition expressed by Equation (1) of the last example is fulfilled. Solution. — Equations (2) in the last example give (x + 3 a) x2 + 3 6a; + c = 0, (x + 2 a) x2 + bx = 0, x2 + 2 ax + 6 = 0. 116 THEOKY OF DETERMINANTS. [chap. v. Eliminating x^ and x from these equations, we obtain (x + 3o) 3 6 c (x + 2a) 6 1 2a 6 = 0, tlie solution of which furnishes the double root. The required value is ^_ 3a(6^-ac)+ c(b-a'^) 2 (&2 _ ac) 10. Find the condition that the equation a^ + 3Hx+G=0 shall have a double root, and find the value of such a root when the condition is fulfilled. Ans. O' + iH^=0; -^ In the equation find the condition : 11. That the three roots may be in arithmetical pro- gression. Ans. 6 a' — 9 a& + 3 c = 0. 12. That the three roots may be in harmonical pro- gression. Ans. 6 6' — 9 ahc + 3 c^ = 0. 13. That the three roots may be in geometrical pro- gression. Ans. 9 a?h^ — c{a? — ah + b^y = 0. 14. That one root may be double another. Ans. 18 (54 a'c - 36 a'b' - 91 abc + 54 6^) -|- 343 c' = 0. 15. That the three roots may have the ratios 1:2:3. Ans. lie — 9a6 = 0. ART. 49.] APPLICATIONS. 117 49. Let it be required to determine whether the two linear factors of a homogeneous quadratic func- tion of two variables are real and unequal, real and equal, or imaginary. Every such function may be written in the form ax- + by- + 2 hxy. Placing this equal to zero and solving for x gives x=-l\_h±{hr-ahy^-]y, which shows that the factors are real and unequal when h''-ab>Q, real and equal when le - ah = 0, and imaginary when h- - ab < 0. The function (^^ — ab') may be expressed as a determinant thus. ■ {h^ - ab) = = D. a h h b The determinant I) is called the discriminant of the given quadratic function, ax^ + by^ + 2 hxy. The required conditions may now be stated as follows : The linear factors of a homogeneous quadratic function of two variables are real and unequal, real and equal, or imaginary, according as the discrimi- 118 THEORY OF DETEKMINANTS. [chap. v. nant of the function is less than, equal to, or greater than zero, respectively/. The same conditions determine the nature of the binomial factors of the complete quadratic function of a single variable, viz. : aaf + 2hx + b, as may readily be seen by making y = 1 in the preceding case. 50. Let us now determine the condition which must be fulfilled in order that a homogeneous quadratic function of three variables may be re- solved into two rational linear factors. Every such function may be written in the form ax^ + by- + cz^ + 2fyz + 2 gxz + 2 lixy. Placing this equal to zero and solving for x gives x=-'^\ (hy + cjz) ±l{h' - ab)f + {cf - ac)z' Qj Jr2{hcj-af)yzf\ (1) If the quadratic function under the radical in the second member of this equation is a perfect square, the given function can be resolved into two rational linear factors. By the preceding article, the condition that the quantity under the radical may be a perfect square is XKT. 50.] APPLICATIONS. 119 {h'-al) {hg-of) = 0. . • ■ (2) (hg-af) ((/- -ac) This equation expresses the required condition. The determinant in the left-hand member may however be transformed into a simpler and more symmetrical form. Thus : {h'- ab) (hg-af) (hg-nf) ig' -ac) a h g (a&-/0 {af-hg) {af-hg) (ac-g-) 1 a a h g ah ab of = a a h g h b / ag af ac (J f c 0. (3) In the same manner we may deduce for the required condition either 0. . (4), (5) a h g = 0, or c a h g h h f h b f (J f c g f c In order that one and all of the conditions (3), (4), and (5) may be fulfilled we must have D = a h 9 h b f 9 f c = 0, • • (6) 120 THEORY OF DETERMINANTS. [chap. v. or we must have c( = 0, 6 = 0, and c = 0, simultaneously. This last supposition reduces the given quadratic function to fxz + gyz + It-xy, which is a prime function of the three variables unless one of the coefficients /, g, or h vanishes along with a, J, and c; but in this case the deter- minant Z) must of necessity vanish. Hence Equation (6) in every case expresses the required condition. The deterjninant B in this article is called th« discriminant of the quadratic function ax' + by^ + cz- + 2fyz + 2gxz + 2Jixy. We may now state the result obtained above in the following form : If the discriminant of a homogeneous quadratic function of three variables vanishes, the function may he resolved into two linear factors. The same condition determines whether or not the complete quadratic function of two variables, viz. : ax' + 2 hxy + hy' + 2 gx + 2fy + c, ART. 50.] APPLICATIONS. 121 may be resolved into two linear factors, as may readily be seen by making 2 = 1 in the case just considered. The student familiar with the calculus may have noticed that the discriminant is the eliminant of the first derivatives of the function to which it pertains, taken with respect to each of its variables. It is a function of great importance in the theory of homogeneous functions or quantics. EXAMPLES. Tell whether the roots of the following quadratic equations are real and unequal, real and equal, or imaginary. 1. 2a?-Zx-^=0. 2. a;^-7a;+3=0. 3. 3a^ + a; + 3=0. 4. 1-^ + 9 = 0. a? X 5. 4a;='-3a; = 0. 6. ~+x + ^ = 0. 7. (a;+3)2+(a;-3)^ = 0. 8. 2a;2-|+10=0. Tell whether or not the following functions are prime. 10. ^-^-'^'^yz-\-2xz-\-xy. 122 THEORY OF DETERMINANTS. [chap. v. 11. 5x''-9y'-5z^ + 18yz + 24:xz-12xy. 12. ax' — bxy + cxz. 33. x''--f + 2y-l. 14. 9y^ + 15yz — 6xz + S xy. 15. ll'x^ + mm'y- + nn'z- + {mn' + m'n)yz + (Zn' + Z'n)a;z + (Z?)i' + Vm)xy. Find the values of A. in order that each of the follow- ing functions may be resolved into linear factors. 16. 12 x^ + \xy + 22/2 + 11 x- by + 2. 17. \xy + 5x + 3y + 2. 18. 2x^--dXf--12z'' + llyz + Xxz-xy. If A, B, C, F, G, H be the co-factors of a, h, c, f, g, h respectively in the discriminant D, then show that :0. 19. B F = 20. A G = 0. 21. A I H F C G C H B 22. A H G H B F G F C = 0. (See Art. 60.) 23. Show that, in order that €13? + &?/2 + cz- + 2fyz + 2 gxz + 2 hxy may be a perfect square, we must have b f = 0, a g = 0, 1 « h = 0. f c 9 c h b ART. 50.] APPLICATIONS. 123 24. By means of tlie preceding examples show that when the function aoi? + hy^ + cz^ + 2fyz + 2gxz + 2 hxy breaks up into two linear factors, then is the function ^„2 + Bv" + Cw" + 2Fvw + 2 Guiv + 2 Huv a perfect square. 25. If {Ix + my + nz) and (I'x + m'y + n'z) are the factors of au? + hy'^ + cz^ + 2fyz + 2 gxz + 2 hxy, we have, by equating coeffi-cients W = a, mm' = h, nn' = c, ??i7i' + m'?i = 2f, In' + I'n = 2g, Im' + I'm — 2 h. Hence, by eliminating I, m, n, V, m', n', from these equations, obtain the discriminant in the form D = abc + 2fgh -af- bg^ - cJi" = 0. 124 THEORY OF DETERMINANTS. [chap. vi. CHAPTER VI. MULTIPLICATION OF DETERMINANTS, AND RECIPROCAL DETERMINANTS. The process of multiplication in determinants and some of the more important of the many inter- esting results to which it leads are considered in the present chapter. 51. Let us assume the two simultaneous equa- tions cCnXi + aioX.2 = 0, 1 in which f \'^) X2 = OioMj -f- 022M2- ^ Substituting these values of x^ and x^ in Equa- tions (1), we have (a,i&ii -I- «io&i2)mi + (aiAi + ai2&22)M2 = 0, 1 (ajl&n + a22&12)Ml + («21&21 + a22&22)M2 = 0. •' These last equations are simultaneously satisfied for values of Mj and u^ other than zero only when (Art. 42) a,i&ii + a^Aii ttifiii + 012622 = 0- ■ (4) *2i6ii -|- cu-fiio 021^21 + f'22622 ART. 51.] MULTIPLICATION OF DETERMINANTS. 125 Now Equations (3) will be satisfied whenever Equations (1) are satisfied, and Equations (1) are satisfied either when or when "u a\2 "21 (1-22 X,= X.2 = 0, (5) 0. But, on account of Equations (2), this last con- dition requiies that hi &22 = 0. .(6) Hence, Equation (4) holds good when either (5) or (6) holds good. It follows that the determi- nants (5) and (6) are factors of the determinant (4), and if this contains any other factor it must appear in every term of the expansion. One term of the expansion of (4) is «iiiii«22^23 ' ^^^ ^^^^ is also a term in the product of the expansions of the determinants in (5) and (6). Therefore On Ou bn 6j2 = 021 ajs hi K ^ii^n "I" <*i2^i2 OuJ,] -|- CI12622 <^21^11 "I" ^22012 O21&2I 4" l „Wl + 62„M2 + ••• + &„„«„• J ■ • (2) The determinant 61. &12 • •61,. 621 622 • ■K 6„i 6„2- -Kn formed from the coefficients of Equations (2), is called the modulus of transformation. When this modulus is unity, the transformation is said to be uni-modular. By what precedes in this chapter it will readily be seen that, if the transformed equations be written PllMl + Pl2«2 ^ 1- i5i„M„ = 0, P21M1 +i'22M2 -I h Pl„U„ = 0, PnHh + Pn^U^ +■■•+ P„„M„ = 0, • -(3) * The transformation from one set of rectangular axis to another in Analytical Geometry is of this sort. Hence the great importance of linear suhstitutions. (See Chap. IX.) ART. 55.] MULTIPLICATION OF DETERMINANTS. 129 then will Pn Pu ■ ■■Ph, = Pii Pk • ■P2n Pnl Pn2- ■Pnn «n ai2 • • «1„ bn 6l2 • •&>„ «21 a22 • • 0-2,, &21 &22 • •62„ a,a «,.2 • ■ a„„ &,.. 6„2- ■ K. that is : i)'' a system of n homogeneous linear' equations in n variables be subjected to linear transformation, the determinant of the coefficients of the transformed equations will be the determinant of the coefficients of the given equations multiplied by the modulus of trans- formation. 55. It has been shown by Professor Sylvester that the product of two determinants of the wth order may, by bordering the determinants as explained in Article 30 before forming the product, be represented in (n + 1) distinct forms. For the case in which m = 3, we have in addition to the form written in Article 51, the following : Writing the determinants in the forms (see Ex. 1, after Art. 30) «!! ^12 "13 and — 6.1 612 &13 a,, «22 ^23 621 &22 &23 «31 a-w ^33 K 632 ^33 1 1 130 THEORY OF DETERMINANTS. [chap. VI. their product is the determinant of the fourth order au&ii + ai2&i2 "ii^ai + '^\^^. o^iAi + «i2&32 ctis '^ifill + W22"J2 %1"21 I 022022 %1"31 "I" C22O32 ^^23 (^sAl + tt32&i2 C3l''21 + C32O22 ^'31^31 + <^32^32 ^^SS 6ig 623 633 Again, writing the given determinants in the forms ftll an ai3 021 «22 O23 %! «32 "33 1 1 and 611 612 &13 621 622 &23 631 &32 633 10 10 their product is the determinant of the fifth order, OillUli Oiyfia ^vfiil C.I2 ftis 021^11 021^21 O21O32 Ct22 CI23 O31O11 O31O21 O-zfi^i (132 (133 612 &22 632 613 623 &33 Once more, writing the determinants in the forms On aj2 0)3 and 1 a,, ^22 O23 1 ttsi ^32 O33 1 1 6u &12 h, 1 K &22 &2a 1 hi &32 K «n 012 ai3 a^i a22 «23 Chl <^32 033 K ^2 K 622 K h. ART. 56.] MULTIPLICATION OF DETERMINANTS. 131 their product is the determinant of the sixth order, &I3 By this method the product of two determinants of the wth order may be expressed as a determinant of any order from the ?ith to the 2?ith inclusive. 56. Formula (1) of Article 53 may be used in forming what may, conventionally, be called the product of two rectangular arrays of equal dimen- sions, such as On aj2 ^ and bn &12 ' Ct2l Ct22 > O21 O22 C'31 %2 '' "31 "32 Representing the result of the operation by A, we have A = anbii + «]26l2 aii&21 + «]2&22 «n&31 + «12&32 = 0- <^21"11 "I" <^22"12 O.21O21 + ft2._;022 O2i0gi -\- d^O^^ <^31^11 + '^320l2 <''31"21 + ^32^22 ^''sfisl + '^32032 This determinant is equal to zero, because it may be regarded as the product of the two determinants 132 THEOKY OF DETERMINANTS. [chap. vi. which we should obtain by adding a column of zeros to one of the given rectangular arrays and a column of elements arbitrarily chosen to the other ; thus : A = ttll an «21 a^i Csi «32 :0. K &12 A: "21 "22 P2S "31 "32 P33 In a similar manner it may be shown that The product (so-called} of any two rectangular arrays having m columns and n rows, in which m < n, is a determinant tvhose value is zero. 57. If the two rectangular arrays are ttll ai2 ttis \ , ?'ll ^2 6l3 > and Oj2\ Ct22 ^23 J "21 "22 "23 the operation of multiplication gives A = O-ifin + «12&12 + «13^13 "11^21 + «12&22 + «I3&23 '''2l''ll + ^22^*12 "r <^23")3 '^2l"21 "I" £'22"22 "I" '*23''23 which may be written in the form A = a,,6,2 + ai3&i3 a,2&22 + ai3623 C'220l2 + «230l3 C'22"22 + ^'23^23 + C^ll"!! + '^13"l3 Ctll021 + '*13''23 ^2\0l\ -\- ft23"13 *21"21 + <'t23"23 ART. 58.] MULTIPLICATION OF DETERMINANTS + 133 '"'SlOll "I" 0!'22^12 <^21"2I "H 0-220^2 '^IS^'ia '*13"23 C'230l3 <^23023 ^12^12 ^12^22 I ai2&12 tt22&22 Each of the first three of these determinants is the product of two determinants, and each of the last three is equal to zero. Hence = a,2 ais . 612 bis + Ctll «13 . 611 bn O22 "23 &22 &23 «2I ^23 621 hi + an «12 ^22 • ^1 &12 622 • This may be generalized as follows : The product (so-called^ of any tivo rectangular arrays having m columns and n rows, in which m > n, is equal to the sum of the m (m — 1) ••• (m — n + 1) n(n-l)-l determinants which may he formed from one of the arrays hy deleting (m — n) columns, each multiplied hy the corresponding determinant formed from the other array. 58. We shall now demonstrate the following theorem, due to Leonard Euler : The product of two numbers, each the sum of four squares, is itself the sum of four squares. 134 THEORY OF DETERMINANTS. [chap. vi. From Example 8, after Article 38, we have a b c d -b a -d c —c d a -b -d —c b a = {d' + b^ + c' + d'y (1) Similarly a /J -13 a -y S -8 -V y 8 -8 y a -/3 ■■{a' + l3' + y'+S'y (2) Let Equations (1) and (2) be multiplied together, member for member. Letting aa + bl3+ cy+ d8 = A, —a/S +ba— cS+dy = B, — ay+b8 + ca — dp=C, — a8—by+cp + da = D, the product of the left-hand members may be written = (A' + S?+C'+D'y^ (3) A B C D -B A -D C -C D A -B -D -C B A ART. 58.] MULTIPLICATION OP DETERMINANTS. 135 H6I1CG (a2 + 62 + c2 + d') {a' + l3' + y' + B') which is the theorem. It may be shown that the product of two numbers, each the sum of four squares, may be expressed as the sum of four squares in forty-eight (48) different ways. The product of n numbers, each the sum of four squares, may be expressed as the sum of four squares in (48)""' different ways. EXAMPLES. 1. Show that the product of any number of deter- minants, of the same or different orders, may be obtained as a determinant of the order which is highest among the factors. Perform the following indicated multiplications : 3. X y 1 X y 1 «! 2/l 1 x„_ 2/2 1 ^2 2/2 1 3^3 2/3 1 a — \ 7i g . a + \ h 9 h h-k f h h + \ f 9 f c-X 9 f c + \ /3 y • /8 7 . a 7 ^ « « ^ y ( J THEORY OF DETERMINANTS. [chap. vi. a h g ■ a g h b f 9 9 f c 136 6. 6. Form the product Xj + Xs Xi + Xs Xi + X2 2/2 + 2/3 2/i + 2/s 2/i + .% 22 +23 2l + 23 2j + 22 and thence show that the first of the two determinants is equal to b f • a g a h f 9 c h b -1 1 1 -1 1 -1 Xi X2 9:3 2/1 2/2 2/3 2l 22 23 m n I 11 I m 8. Obtain the quotient, — ma + nb mb + nc —mc + na la + nb — lb + nc Ic + na la —ma — lb +mb Ic —mc in the form of a determinant, and thence show that the first determinant equals zero. 9. Find two determinants whose product is ax + CX al + en ay + cy hm bm cx +ax d + an cy + ay ART. 58.] MULTIPLICATION OF DETERMINANTS. 137 10. Prove the equality : «i" + &i + Ci ajfta + 62 + Ca ChO-a + &3 + C3 62*^3 + C3 ^-3 62&1 + Ci V + C2 C3C1 C3C2 tti 62 as &3 (See Eq. 3, Art. 29.) 11. Show by means of the multiplication theorem that tti 61 Ci di 2 ^2 ^2 "2 , 0<> Co Uq ^4 ^4 d. ai + bi + Ci Ui + bi + di ai + Ci + dj &i + Ci+di a2 + b« + C2 «2 + &2 + CZ2 «2 + C2 + <^2 &2 + C2 + ^2 a3 + &3 + C3 a3 + &3 + <^3 tts + Ca + cZa 63 + Cj + dj Wl + ^l + Ci a4+&4 + d4 «4+C4 + d4 64 + 04+^4 12. Generalize the preceding example. (See Ex. 24, after Art. 28.) 13. By means of the product, a + 6V — 1 — c+dV— 1 c + dV^ a — &V^^ « + ;8V-l -y+SV-l y + SV^ «-j8V^ show that the product of two numbers, each the sum of four squares, is itself the sum of four squares. 138 THEORY OF DETERMINANTS. [chap. vi. 14. By writing each factor as a determinant and performing the indicated multiplication, show that the product, is the sum of two squares. 15. Express the product of two determinants of the second order in three distinct forms. (See Art. 55.) Reciprocal Determinants, 59. If the elements of a determinant are replaced by their respective co-factors, the new determinant thus formed is called the reciprocal of the given determinant. Thus, the reciprocal of S = Ct^i ^li CE-nl ^'n2 (1) is the determinant (2) A^ in A being the co-factor of the element a^ in S. The reciprocal of a determinant is also called its determinant adjugate. ART. 61.] EBCIPKOCAL DETERMINANTS. 139 60. Theorem. — The reciprocal of any determinant of the nth order is equal to the (n — l~)th power of the given determinant. This may be shown as follows : Multiplying together Equations (1) and (2) of the last article, member for member, we have «2ij4iiH hao^Ain ' a2i^2iH \-a2nA-2„ ■••a2iXiH i-a2„A„„ flnl^uH hCtnnAln a„1^2lH h a„„^2„ •■• a„lA„\-^ hann-Ann By means of the formulse in Articles 27 and 28, the above equation may be reduced to :S", or 8.A = 8 0- • 8- • , 0- ■ 8 A = 8"-', ich is the theore m. 61. Theorem. — Any minor of the vath order in the reciprocal of a given determinant is equal to the product of the cof actor of the corresponding minor in the given determinant by the (m — lyth power of the given determinant. Assuming 8=1 «!! a^s ■■■ a„„ I 140 THEORY OF DETERMINANTS. [chap. VI. as the given determinant, let the elements of the minor be taken from the /th, c/th, hth, ••• rows and from the pth, qth, rth, ••• columns of the reciprocal determinant A=| An A,2 — A„„ |. Also let f+g + 7i+ ...+p + q + r+-:=u. The minor in question, which we assume to be of the TOth order, may be written in the form (Art. 30) A„ = -"* -^/« -"-fr ' ' ' -"■/, m+1 ■ ■ ■ -"-/n Agp A, A,r- •• -^jr.m+l • ■4 A,, A,, Ajtr • • -^ft, m+1 • ■ A • • 1 •0 • • • 1 (1) In the given determinant let us pass the above numbered rows upwards and the above numbered columns to the left. In this manner we obtain (Art. 35) 8 = (-l)" Ofp dfq O, r ••• «/,„+! ■• a. 7f, m+l '*m+l, p <^in+l, ? '■'ni+1,, "" C'm+l, m+1 "■■ ("m+l, ?i ..(2) AKT. 62.] RECIPROCAL DETERMINANTS. 141 Multiplying together Equations (1) and (2), mem- ber for member, gives A„-S = (-l)" s ...0 ...0 s ...0 .".. 8 ...0 ... V, m+1 ^'j, m+1 Cf/i, in+l •" f'm+l, m+1 "-n, m+l 'fit ■'»n ■*■ "m+I, n :(-l)"S" a, 'm+l, ffl+1 ' ' ' *-^n, m+l •^m+J, n , or A„= 8-H-l)" a„+i,„+i ••• «„+!,„ . (3) which establishes the theorem, since the minor of S in the second member is, with the sign-factor ( — 1)", the co-factor of the minor corresponding to A„. 62. If S = 0, Equation (3) of the last article gives, for all values of m greater than unity, A„=0. That is. If a determinant is equal to zero, all the minors of its reciprocal which are of an order higher than the first are also equal to zero. (See Examples 19, 20, and 21, after Article 50.) 142 THEORY OF DETERMINANTS. [chap. vi. If m = 2, we have for A„ any one of the following : • • = 0. A A A A ai 9r Then, Ap '■ -^gp '■'■ -^fi'- -^gq '■ '■ ^/r- -^gr '■■••• That is, if a determinant is equal to zero^the co- factors of the elements of any row are proportional to the cof actors of the corresponding elements of any other row. (See Art. 43.) 63. The theorem in Article 61 gives, employing the notation explained in Article 36, (1) ^fp A = 8 co- «/. «/, Ap ■^gi agp a.. or, in the notation of the calculus (Art. 89), = 8. d?h daj^da^,' dS_ dS^ da,^ da,, d8 dS da„„ da,„ whence, by expansion, g d'8 ^ dS d8 dS dh daj-^da^j da^j, ' da^, da^j, ' da^,' (2) ART. 64.] DETERMINANTS OF SPECIAL FORMS. 143 CHAPTER VII. DETERMINANTS OP SPECIAL FORMS. There are certain classes of determinants which, by virtue of some mutual relation among the elements, or of some particular disposition of the same, possess properties peculiar to themselves. In the present chapter a few of the more important of these special forms of determinants will be examined. Symmetrical Determinants. 64. Two elements of a determinant so situated that the row and column numbers of one are respectively the column and row numbers of the other are called conjugate elements ; thus, in the determinant | a-^^ a^^ ••• «„„ |, the two elements a^^ and a^ are conjugate to each other. Each element of the principal diagonal must be regarded as its own cor{jugate. In the same manner we may speak of conjugate co-factors of a determinant ; A,^ and A^, being such in reference to the determinant, just written. 144 THEORY OP DETERMINANTS. [chap. vii. The terra conjugate must not, in this connection, be confounded with the term complementary. A row and a column having the same number are sometimes called conjugate lines. 65. If each element of a determinant is equal to its conjugate the determinant is said to be axi-symmetric ; thus, the determinant a a b c a /3 I m b I y n c m M' S is axi-symmetric. The principal diagonal is called the axis of symmetry. If the elements of the axis are zeros the determinant is said to be zero-axial. If, in I a^^a^ ■■■ a„„ |, (^ks = 0-7l-a+l, n-k+1 ) the determinant is symmetrical with respect to the secondary diagonal. Such a determinant is z y X 1 p X r P y r Q z which is also zero-axial. Any determinant in this ART. 67.] SYMMETRICAL DETERMINANTS. 145 form may be transformed to one which is sym- metrical with respect to the principal diagonal by simply reversing the order of the columns (or rows), the sign being changed unless, n being the order of the determinant, either n or (« — 1) is divisible hy four. The following properties of axi-symmetric deter- minants may be deduced directly from the definition: (a) Conjugate lines are alike. (b) Conjugate minors are equal. (c) Minors which are co-axial with the given deter- minant are axi-symmetric. (Ay The reciprocal determinant is axi-symmetric. 66. Theorem. — The square of any determinant is an axi-symmetric determinant. This follows directly from Formula (2) of Article 53. For if a^ = K for all values of r and I from 1 to n, the formula referred to gives ft. = a*ia.i + «*2a.2 + • • • + «t A„ = a.Ai + a.2«*2 H + «,„«*„ =ft« which establishes the theorem. 67. Theorem. — If an axi-symmetric determinant be multiplied by the squai-e of any determinant of the same order the product will be an axi-symmetric deter- minant. 146 THEORY OF DETERMINANTS. [chap. vii. Let I a-iia^i '" <^nn \ be the given axi-symmetiic determinant, and ( iiiJ22 "■ ^"n | the determinant by the square of which it is to be multiplied. Also let I hi K ■•■ Kn i • I ttll «22 ••• a.m |=| Pll Pn — Pnn \- We are to prove that, if I ^11 -P22 ••• Pn,> 1 = 1 Pn P22 ••• Pnn i ' I ^11 hi -■ Kn \, then is | PuP^i ••• Pnn \ axi-symmetric ; that is, P — P By Formula (2) of Article 63, we have Pj,> = PkAi + Pmb.2 H h PtAn = (Mil + Ml2 -I f- Kn«ln)b,i + (M21 + M22 H 1- &*«a2„)6.2 + (&««»! + M„2 H h b„„a„„)b,„, or, since a^i = ai, Pk. = (&.i«ii + &.2a2i -\ h 6.Ai)&H + (^ia2i + V'22 H h M2„)&*2 + (M„i + &.2an2 H 1- &s„a„„)6in = P.Al +i'A2 H \-PsAn=P.k, which establishes the theorem. ART. 69.] SYMMETRICAL DETERMINANTS. 147 68. Corollary. — All powers of an axi-synimetric determinant, and the even powers of any determinant, are axi-symmetric determinants. This readily follows from the theorems demon- strated in the last two articles. EXAMPLE. Tell whether or not the product of two axi-sym- metric determinants of the same order is axi-symmetric. 69. If all the elements on every line perpendicular to a diagonal of a determinant are equal, the deter- minant is said to be per-symmetric or ortho-symmetric ; thus, the determinant A = "I tta C63 ... a„ a^ «3 a^ ••• «„+i a-i a^ «5 ■ • ■ "n+S (1) ''n "'a+1 "n+2 ■" "2ii-l is per-symmetric with respect to its principal diag- onal. The whole number of different elements in the above array being (2n — 1), we may write the per-symmetric determinant of the nth order in the abbreviated form P(ai, a,, ••• a2„_i), . ... (2) which is identical with (1). 148 THEOKY OP DETERMINANTS. [chap. vii. 70. Let the elements of the per-syrametric deter- minant given in the last article be written down seriatim, and the successive difference series formed ; thus: ttj — ai tts — 02 a^ — 03 ttj — 2a2+aj a4 — 2a3 + a2 a4 — 3 03 + 3 aj — ttj Introducing an obvious system of abbreviated notation, the above array may be written Oi ttj ctj a^ A^'% AWas AWttj AC'ai We shall now demonstrate the following 71. Theorem. — The per-symmetric determinant P(aj, aj, ••• a2n-i) is equal to the per-symmetric deter- minant P(ai, Aai, ■•• A'''--^' a^ . Replacing each element of the determinant A of Article 69 by that element minus the corresponding element in the preceding row ; in the determinant AKT. 72.] SYMMETRICAL DETERMINANTS. 149 thus formed, replacing each element of each row after the second by that element minus the corre- sponding one in the preceding row; and continu- ing this process till finally the elements of the last row are the only ones operated upon, the resulting determinant may be written (Art. 22) A<"a, ••• A<"a„ A'^'a, A<2)a.. A<"-"ai A<"-''a2 A<"-%3 ••• A(»-%„ (1) Operating in precisely the same manner upon the columns of the determinant (1), we have finally A= tti A<%i A'2'ai ••• A("-"ai !, . .'(2) A'l'ai A'^'tti A'^ai - A^"% A<2)ai A<-'')ai A<«ai ••• A<»+"ai A<"-%i A<"'a, A("+% ••• A<2-2)ai which we were to prove. It will be observed that the determinant (2) is also per-symmetric. 72. If each row of a determinant may be derived from the preceding row by passing the last element 150 THEORY OF DETEEMINANTS. [chap. vii. over all the others to the first place, the determinant is called a circulant ; thus, .... (1) A = «i a,. a. «i is a circulant. The notation A=C(ai, tto, ■■■ a„) ■ (2) is very generally employed. Circulants are obviously per-symmetric deter- minants, but they possess some properties not possessed by per-symmetric determinants in general. The most important of these is stated in the fol- lowing 73. Theorem. — The determinant C (aj, a2, •■• a„) contains as a factor a^^ + a2r + agr^ H |-anr"~\ in which r is one of the roots of the equation x„ — 1 = 0. Let us consider the product (ai+ a2?-+a37-2-l- • • ■ +o„r"-i)(^i-^^2»-~'+^3''~^+ • • • +^„r-"+i) . For the coefficient of r*"^ in the above product we have, remembering that r'~' = r"+'~' , ART. 75.] SKEW DETEKMINANTS. 151 When k=l, this coefficient becomes a^Ai + a,^2 -\ h "nA, which is equal to the given determinant ; but for all other values of k it vanishes (Arts. 27 and 28). Hence the given determinant contains the factor a I + a,?' + ay- -\ \- aj-"'^ . 74. If we let (j>{r) =«i + a^r H h a„r— ', and represent the different roots of a;" — 1 = by rj, r^, ■•■ r„, we may, by the preceding article, write 6'(ai, fla ••• an)=<^(''i)-'^('-2)-"<^(0- Since one of the values of r is unity, (?(«!, a,, •■■ a„) is divisible by a^+ a^-i |-a„, that is: A circulant is divisible hy the sum of its elements. Let the student show this by another method. Shew Determinants. 75. If each element on one side of the diagonal of a determinant is equal to its conjugate with its sign changed, the determinant is called a skew determinant; thus. 152 THEORY OF DETERMINANTS. [chap. VII. A = « a b c . • . . .(1) —a P I m -b -I y n — c — m -n S is a skew determinant. If every element of a determinant is equal to its conjugate with its sign changed, the determinant is defined as skew-symmetric. Since the elements on the principal diagonal are their own conjugates, it follows that skew-symmetric determinants are zero-axial. The determinant (2) A = a b c —a I m -b -I n — c —m —n is skew-symmet ric. 76. The following properties of skew-symmetric determinants correspond to those of axi-symmetric determinants given in Article 65 : (a) Conjugate lines differ only in the signs of their elements. (b) Conjugate minors are equal, or differ only in sign. (c) Minors which are co-axial ivith the given deter- minant are themselves skew-symmetric. ART. 77.] SKEW DETERMINANTS. 153 (d) The reciprocal determinant is skew when of even order, hut axi-symmetric when of odd order. This last property may be proved as follows : Let the given determinant be | a,i a^^ ••• «„„ |, and its reciprocal | A^^ A^.^ ••• J.„„ |. The co-factor A^^ differs from the co-factor A,^ in the sign of every element ; hence, -4*8= ( — 1)"~-4«J:. If n is even, A^= — J..^, but if n is odd, A^^ = A,„; or, the reciprocal determinant is skew when n is even, axi-symmetric when n is odd. 77, Theorem. — A skew-symmetric determinant of odd order is equal to zero. For, multiplying each element by —1 simply changes columns to rows and rows to columns, which does not change the value of the determinant. Hence, letting S be the given determinant and w its order, S= (-1)"8, which, if n is odd, can only be true when 8 = 0. EXAMPLE. Show that the reciprocal of a skew-symmetric deter- minant of even order is not only skew, but skew- symmetric. 154 THEOEY OF DETERMINANTS. [chap. VII. 78. Theorem. — A sTcew-symmetric determinant of even order is the square of some rational function of its elements. Let us represent the given skew-symmetric deter- minant, which we assume to be of the 2wth order, by A. Also, let «„ be any element of the given determinant and A^, the corresponding element of its reciprocal. Then, by Equation (1) of Article 63, we have ■ • (1) A„ A, = A co- «rr flrf Air Al «;, a„ But, since A is skew-symmetric. o„ = = a„. A,= -A :0=A, (2) By means of (2) we may write Equation (1) in the form {A^iY A = CO- a,, -an (3) The denominator in the second member of the above equation is, by Article 76 (c), a skew-sym- metric determinant of the order (2?i — 2), and, by Article 85, its sign factor is ( — 1)2<'-+') or -|-1. AKT. 79.] PPAPFIANS. 155 Since A is, by Equation (3), a perfect square when the denominator of the second member is such, it follows that if any skew-symmetric deter- minant of even order is a perfect square, that of the second higher order must also be a perfect square. Now, any skew-symmetric determinant of the second order is of the form e -e which is a perfect square. Hence, by induction, anj' skew-symmetric determinant of even order is a perfect square ; which is the theorem. As an illustration, let us resume the determinant (2) of Article 75. By Equation (3), making r = 4 and 1 = 1, this determinant may be written a b c I m -I n A = I -I A=(a?i — bm + ciy. . or Pfaffians. 79. The last article introduces us to a class of functions known as Pfaffians, which we now proceed to examine. 156 THEOKY OF DETERMINANTS. [chap. vn. In the skew-symmetric determinant of even order A= «12 ^13 ai4 (^l, 2n-l O'l, 2n — a^ "23 a-ii <*2, 2n-l ''2, 2n -ai3 — ^23 «34 ^(■3, 2n-l C3, 2» -an —a^i -"34 <*4, 2>i-l Q4, 2n — Cf], 2n-l — '*2, 2n-l — '"s, 2n-l ^4, 2ii-l "■ " '*2n-l, 2i — '^l, 2™ — ''2, 2(1 — '*3, 2n "'4, 2n ' " " ^2n-i, 2» " (1) one of the terms is a\.ji 34a 50 • ■ • « 2„_i, 2,1 (2) Since the determinant A is a perfect square, one of its square roots contains the term + <^12'^34 (") while the other contains the term — «12'*34'''j6 ••• '*2;i-l,2n (4) That square root of A which contains the positive term (3) is called the Pfaffian* of the elements above the zero axis. Thus, in the preceding article, an — hm -f cl * These functions were named by Professor Cayley after the mathematician Jean-Frfiderio Pfaff (1765-1825), who proposed a problem of historic interest in the discussion of which they were used by Jacobi. ART. 80.] PPAFPIANS. 157 is the PfafBan of the elements a, b, c, I, m, n. 80. The Pfaffian related to the determinant (1) in the preceding article may be represented by the array P=\ tt], a]3 an 023 «24 <^1, 2n-i (h <^4 • • • f -i, 2n-l ■2,271 ^3,2)1 ••■Sji-", 2n-l "2n-2, 2>i Oan. (1) However, the umbral notation P= I «12 <*23 "34 ■•• *^2n-l, 2n |) or F=ff{(h3. «ffl «« ••• aii-i,2n) (2) (3) is more convenient. These are still further abbrevi- ated to P= ij a,,2„ I and P=ff{a,,^), . . • (4), (5) respectively. The term is called the principal term of the above Pfaffian. 158 THEOEY OP DETERMINANTS. [chap. vii. 81. The order of a Pfaffian is the degree of its terms in reference to its elements. Since a Pfaffian is the square root of a skew-symmetric determinant, its order is one half that of the determinant to which it is thus related.* Thus the Pfaffian P of the preceding article is of the nth. order. The definitions of the terms column, row, and element as applied to determinants answer equally well for Pfaffians. As in determinants, any element may be defined by its row and column numbers, but this method is usually modified as follows : The first row of a Pfaffian of the nth order contains (2n — V) elements and is called the first frame line. The line through the first column and the second row also contains (2n — 1) elements and is called the second frame line. Similarly, the lines through the second column and the third row, the third column and the fourth row, ..., and through the last column, contain (2ji — 1) elements each, and are called respectively the third, fourth, . . . , and 2wth frame lines. The position of any element is defined by giving the numbers of the two frame lines intersecting in the element in question. * Some writers regard the order of a Pfaffian as being the same as that of tlie corresponding skew-symmetric determi- nant. ART. 83.] PFAFFIANS. 159 This system is illustrated, by the following array : ■12 i«5r -^ 1st frame line. -^ 2d " ■* 3d " " ■^ 4.th « " It should be noticed that the element in the Zth and rth frame lines of a Pfaffian is the same as the element in the Ith row and the rth column (li " '^t. 2n-I (1) — (^2,2n-l — "'■3,211-: Deleting the first row and the last column of the determinant just written, leaves us A(l,2,._ '*(2ii, 1 = «34 a. '3, 2ji-l (2) — ^2, 2n-2 — %, 2n-3 — f*!, 2n-2 "•" *2n-2, 2>i-I — Ct2_ 2n-l — ''3, 2n-l — Ct4, 2n-l""" which is a skew-symmetric determinant of even order. By Equation (3) of Article 78 we have {AiLr = A.ASkfl. ..... (3) ART. 83.] PFAFFIANS. 161 Taking the square root of both members of this equation gives ftjs ft24 ■ ■ ■ "'2, 2ii-l «34 • • • O'J, 2n-l ^/■2n = 1 ^1-i Ctl3 ^li ■■■ 0^1,2). • I <^'23 «24 " " " *2, 2ii-l ) • -(4) (J23 ^24 • • • Cl,2^ 2)1 %4 ■■■ '^S, 2>i ''2>i-l, 2.1 which establishes the theorem for the case in which the given determinant is of odd order. (6) In case the given bordered determinant is of even order, let us consider it as derived from the skew-symmetric determinant A' = ai2 ttia ai, 2„-i — ai2 «23 «2, 2«-l — ftis — a23 • aa, 2„-i — %, 2n-l —'^2, 2n-i — '"s, 2n-l • • • 1 ■ -1 ,(5) by deleting the first column and the last row, and then reducing by Article 29. This gives , . . .(6) ^ (2™ — ai2 -023 a. 13 a, a^s •■• 0,2, 2n-l ••• «3,2»-l — ^2, 2n-2 — O's, 2n-2'" f'2n-2, 2n-l 162 THEORY OP DETERMINANTS. [chap. vii. which is a bordered skew-symmetric determinant of even order. Since A' is skew-symmetric, we have, by Equation (3) of Article 78, JA'jLr = A'.A'|kf'; (7) Now let A represent the skew-symmetric determi- nant A' III, which is of odd order (2n — 1). Then A' = A' <^;;; ^-1 = ASi;;i}. (See Art. 29. ) I ... (8) Substituting these values in Equation (7) gives {AlL-.r = A4 • • • (mo %n-2, 2ii-l >(10) which completes the demonstration of the theorem, since AJ2„_i is a bordered skew-symmetric determi- nant and is of even order. ART. 84.] PFAFFIANS. 163 84. Equation (4) of the last article may be written ASJ„ = -P-P(2,.,i, (1) in which A is a skew-symmetric determinant of even order 2n, and P its Pfaffian. Also, Equation (10) of the same article may be written ^(2n-l = -P(2ii-1 -P (1 ) (-) in which A is a skew-symmetric determinant of odd order, P,2„_i the Pfaffian of A[i"_}, and P^l the Pfaffian of A'l. Since Equations (1) and (2) depend upon the general Equation (3) of Article 78, they may them- selves be generalized, giving respectively K=P-P^.,. (3) when A is of even order, and K=P P(\w •■• -P(i,2. may be ex- panded in the same manner as the original Pfaffian. We thus have a method of forming the complete expansion of a Pfaffian in terms of its elements analogous to the method of expanding determinants explained in Article 27. The method is illustrated in the development of the following Pfaffian : — a p=\ a b c d e f 9 h i J k I m n j k I -b 1 g h i + c 1 / h i m n m n k I 166 6 THEORY OF DETERMINANTS. [chap. VII. -d \ f g i + e !/ 9- h J I J k n m = a{jo - - kn + lm)—b (go — hn + im) + c (/o - hi + ik) -d{fn-gl+ij)+e{fm -gk + hj). 86. An inspection of the method of expansion explained in the preceding article shows that the number of terms in the complete expansion of a Pfaffian of the nth order is the product 1 • 3 • 6 (2n-l), an odd number. Of these terms there is one more positive than negative. EXAMPLES. 1. Expand \ a b c d e f g h i and explain the significance of the result. 2. Prove that if any two frame lines of a Pfaf&an be interchanged and the element at their intersection changed in sign, the value of the Pfaffian is changed only in sign. State and prove a property of Pfaffians analogous to t.hat established for determinants in ART. 87.] ALTERNANTS. 167 3. Article 20. 5. Article 61. 4. Article 60. 6. Article 62. 7. Show that a Pfaffian, one of whose frame lines contains only binomial elements, may be expressed as the sum of two PfafBans. 8. By squaring both members of the result in the last example prove the theorem of Article 83. 9. Prove that "(r • "(s ■ • ■' (r, i • J (s, SJ A being a skew-symmetric determinant of even order of which the corresponding Pfaffian is P. 10. Prove that if an axi-symmetric determinant van- ishes, the co-factor of any element is a mean propor- tional between the co-factors of the principal diagonal elements belonging to the row and column of the given element. 11. Prove that, if the minor of the leading element of an axi-symmetric determinant is zero, the determinant is expressible as a second power. 12. Prove that the co-factor of the sum of the ele- ments of a circulant of the order (2?i — 1) is expressible as a determinant of the nth order. (See Art. 74.) Alternants. 87. A function of two or more variables such that the interchange of any two of the variables 168 THEOEY OF DETERMINANTS. [chap. vii. changes the sign without changing the value of the function is called an alternating function of these variables. Alternating functions are to be distinguished from symmetric functions^ in which the interchange of two variables changes neither the sign nor the value of the function. As examples of alternating func- tions we give : {x-yY, x^y — xy- -\- xz'- — 3?z -\- yH — yz', sin(x^ — y'^), X sin y —y sin x, X and log: y On the other hand, i^-yy, x- + y- + z- + yz + xz + xy, cos(a^-i/2), X sin y + y sin x, and log a;]/ are symmetrical. 88. The determinant Xl e^i sin Xi X2 e^2 sin 0:2 x^ gX, sin ajj ART. 89.] ALTERNANTS. 169 is an alternating function of x^, x^, and 2:3, for the reason that interchanging any two of these variables amounts to the same thing as interchanging two rows of the above determinant, a process which only reverses the sign. Such a determinant as the above is called an alternant. An alternant may be defined as a determinant in which the elements of the first row are functions of some variable, the corresponding elements of the second row the same functions of another variable, etc. Any alternant whose elements are functions each of a single variable may be expressed as follows : A = /o(a;i) /i(a;i) (1) for which has been adopted the notation A = Alfo{xO, /i(xo), •■•/„. i(a;„)]. A simple alternant is one in which the functions used as elements are powers of the variables. 89. Let us compare the simple alternant /J /«. « « 2 « n— 1\ -:- Xi" Xn n n 170 THEORY OP DETEEMINANTS. [chap. vii. with the product P={X2-Xi)(Xs-Xi)(x^-Xj) ... (X„-Xi) X (% - x-j) (a;4 - ajj) ... {x„-x.2) X{x^-Xs) ... {X„-Xs) X (a;„ - a;„_,) ; that is, the product of the differences obtained by- subtracting each of the n quantities a^j, x^, ••• x^ from each of those which follow it. The determinant vanishes upon any two of the quantities, as Xi and a;^, becoming equal, and is there- fore divisible by {x^ — x^. Being thus divisible by any of the factors of P, the determinant is divisible by P. Since the determinant and P are of the same degree, -z (w — 1), with respect to the quantities Xy, ^2, ••■ a^ni their quotient must be independent of these quantities. To obtain this quotient we may compare any term of the expanded product with that term of the expansion of the determinant in which Xy, x^, ■■■ x„ are affected by the same expo- nents. The product of the first terras of all the binomial factors of P is 7* 'y. T* ^ but this is also the leading term of the determinant. Hence, the quotient sought is unity, and A{x^^, X,, X.?, ■■■ «/ ') = P. ART. 91.] ALTERNANTS. 171 90. The product P is called the difference product of a^j, x^i •■• x„, and for it the notation P = ^\xi, x^, ■•• a;„) has been adopted; in accordance with which the notation for the square of the difference product. The alternant A (x^, x^j, x^, ■•■ a;„""') is called the difference product alternant. 91. In the alternant A [/oCa^i), fi(x^), ■■•/«-iC'Cn)] let fi(x) be, in each case, a rational integral function of the degree i; and a, the coefficient of x\ the highest power of x iwfjQc). Multiplying the elements of the first column, each of which must be represented by «(,, by the proper factor and subtracting from the correspond- ing elements of the second column, we have a new second column containing the elements a^x^^ o-iX^i ■ ■■ «ia-„. Then, subtracting the elements of the first and second columns, each multiplied by the proper factor, from the corresponding elements of the third column, we have a new third column made up of the elements a^x-^, ^2^2^' "■ ^-i^n- Continuing this process, we finally reduce the 172 THEORY OP DETERMINANTS. [chap. vii. given alternant, without change of value (Art. 22), to the form ««-!»!" (1) Hence, by the last article, ^M^MMh^:iA=M)l^a,a,a, ... a„_,. . .(2) This result is a special case of the theorem in the next article. 92. Theorem. — Every alternant whose elements are functions of x^, Xj, ••• x^ is exactly divisible by f*(Xj, Xj, ••• x„), and the quotient is a symmetric function of these quantities. For, the alternant vanishes upon any two of the quantities x^, x^, ••• x„, as Xt and x^, becoming equal, and therefore contains the factor Xt — Xf, and conse- quently the product of all such factors, which is f'(a;j, ^2, ••• 2;„). Moreover, since the interchange of Xi and x^ changes the sign, both of the alternant and of f*(a;i, x^, •■■ x„), the quotient remains unaf- fected by any such interchange and is therefore a symmetric function. ART. 93.] ALTEKNANTS. 173 93. We may express the coefficients of the rational integral function of a; as symmetrical functions of the roots in the following manner : If rj, r^, ... r„ are the roots of such a function of X, we have of + ajK"-' + (uc"-- -\ h a„ 1* + o„ = {x-r,){x-r^) ■■■ (x- ?■„) = 0. (1) Multiplying both members of the above equation by ? (»'ii ''21 ■•• '"«)» the result may be written ^^('•i, r„- r„) (x" + a,x"-' + a-jx'-' +... + «„) 1 n n^ 1 rj u- 1 r n ' n (2) la; a^ ■■• of Developing the determinant according to the ele- ments of the last row and equating the coefficients of the same powers of x in the first and third members gives : a„ = (-1)' ^-('V '-2, •••'•») •■• ro" ' n ' w 174 THEORY OF DETERMINANTS. [chap. tii. _ (-1)"-^ ^^('•i, n, ■•■ 7-,) 1 9f ... 9-i" 1 r/ ... r," 1 r ' (-1)"-^ ^H'"i) '■» ••• '•«) 1 1- ...■)• *-l r *+l tti (-1) 1 n ■■■ j'l''-^ ri" 1 r. ■■■ r,"-' n" ■*■ ' n ' n ' n The above functions of the roots are symmetric by Article 92. 94. Resuming Equation (1) of the preceding article, we have, by the theory of equations : «„ =(-l)"'V2 ••• '■„; «„ , = (-l)"-'2( + v, ... n); n-l and, in general, n„_, = (-l)-'S(+v-, -n), ri-t in which S implies that the sum is to be taken of 91— i all terms of degree (n — lc) that can be formed from ART. 95.] ALTERNANTS. 175 rj, rj, •■• r„, no root to appear as a factor more than once in the same term. Comparing the above value of a„_t with that given in the preceding article, we readily obtain the general relation 1 1 1 X,- .x," 1 X2 ■ ■ a!/-i a;/+i xi+^ ■ ■x^ 1 x„- a;„'+i ^11 • a;/ 1 a;i ••• a;/"' Io* ... 1* * — ^ J^2 '^2 a;/ 1 />* . . . O* *~^ /Yi ft rf *+l n. 11 - 1 -L J/„ • • • X„ J,„ J,„ -^n = 2(+a;ja;,.-.a;,). 95. Squaring the difference product %^-(ay, a^^ ••■ a„) by Formula (1) of Article 53, we have, repre- senting a^' + ttj* + ■•• + a,l by Sj, So Si S2 • •■ S„_i Si Sj S3 • •• s„ S2 S3 S4 • •• S„+i Sn lS„ S.+1 • •• S2»-2 176 THEORY OF DETERMINANTS. [chap. vii. EXAMPLES. 1. Prove the result in Article 94, without reference to the theory of equations. = — ^H«D «2) «3)> 2. Prove that 1 Ui+Qs a^a^ 1 ttj + fl'i %«i 1 tti + flj <^l«2 and give the corresponding expression for ^^{(h,a2,n,i,a^. 3. Prove that -3^^ («!, a^, a^, a^) = | (a^ - a3)'(a4 - a2)'(a4 — "i)^ (ttj — a2)Hct3-ai)^ (aj - ai)3 and give the general theorem. 4. Pind the co-factor of ^^ {a„ ••• a„) ^^ (6i, ■■■ b„) (% - &i)"-' (ai - 62)"-^ - («i - Ky-' {a, - b,y-' {a, - hy~' ... (a, - &„)»-i (a,. - 6,)''-i (a„ - 62)""' - (a» - KY" 6. Find the co-factor of ^* (aj, .•• a.„) ^^ (6), •..&„) (a, -6i)--' (aj-ft^)-'... (ai-6„)-i {a„-b,)-' (a„ _ 6,) -1... (a„ _&„)-' ART. 95.] 6. Show that 1 Qi tti' 1 02 a2^ 1 as a-' ,3 ALTEKNANTS. -=- ^^ (cti, a^, as) 177 + ■ a^" ■ + - («i-a2)(ai— as) (aj— ai)(a2— aj) (as— aO (as— «2) = aj + ttj + tts. 7. Show that 1 ai ••• tti"-^ Oi' 1 02 ••■ tta""^ aj' -=-^^(ai, aj, ••• a„) + + + + + + (tti — a.j) (ai — as) ••• (Oi — a„) a/ (a2 — ai) (02 — tts) • • • (aj — a„) a/ («r — ttl) (flr — ^2) •• ■ ("r " «r-l) («, — a,+ i) " " («r " «J Ojj.! («„-i - ai) • ■ • (a„_i — a„_2) (a„_i — a„) Oj («„ - en) • • • (a« - a,-2) («n - «n_i) ' Suggestion. — Expand the given determinant dividend accord- ing to the elements of the last column. 178 THEORY OF DETERMINANTS. [chap. vii. 8. By the aid of the preceding example show that ^hai, 0.2, ■■■ a,.) C^(«i) «2, ••• «„) + 9. Show that ^^(tti, as, ■•• a„) is the complete symmetric function of degree (g— n+1) of the 71 quantities ay a^, •■■ a„. 10. As a special case of the above show that 1 a; a;' -=- ^• (a;, y, z) i y f 1 z z^ = a^ + 2/^ + ^^ + 2/^2 + 2/2^ + ^^^ + a-"2^ + ^y + a^y^ + ^2/*- Continuants. 96. A determinant whose elements are all zero except those of the principal diagonal and the two adjacent minor diagonals, and in which the elements of one of these minor diagonals are all —1, is called a continuant. Thus ART. 96.] Qn = CONTINUANTS. 179 «! -1 ...0 ... a) &2 aj -1 ...0 &3 a,--0 ... a„ 1 -1 - K a„ is a continuant of the nth order. By an orderly cliange of columns into rows, we have Q.= «1 h .. 1 a^ h- -1 ttg.. .. a.,- b, .. -1 a (2) which shows that we may interchange the two minor diagonals. Again, by reversing the order of both columns and rows, we may write Q.= a„ -1 ... K ttn-l -1 ... &„-! a»-2 ... •••0,2 -1 -h 0.1 (3) 180 THEORY OP DETEKMINANTS. [chap. vii. that is, the order of the elements in the diagonals may be reversed. The notation usually employed for continuants is as follows : ^/ 6, 63 •■• &„_i h„ \ 97. Expanding the second member of Equation (1) of the last article with reference to the elements of the last row, we have &2 h •■■ K-i \ , i f ^2 h ■•• K-2 ^" "''\ai a^ as •■■ a„_J '\ai a.^ a^ ■■■ a„_2 = a,M.-i + KQn-2 (1) This equation affords a convenient method of writing the expansion of a continuant in terms of minors co-axial with itself. Thus 0,1 a^ Og aJ \ai a^ aJ \ai a^j = a,al ^2 ']+afi,{a,)+hI ^' \ = aia^a-itti + 04^362 + «4ai&3 + a-flfii + hfi^- 98. If M„ is the number of terms in the expansion of the continuant of the order n. Equation (1) of the last article gives M„=M„_i+M„_2, (1) a difference equation, the solution of which is ^ ^ (l+V5)"+^-(l_-V5)''+^ ,2) 2"+V5 AKT. 98.] CONTINUANTS. 181 This is an integer for every integral value of n, as may be shown by the binomial theorem. Note. — The reader who is not familiar with the Calculus of Finite Differences may deduce Equation (2) by the following somewhat arbitrary method: Let Equation (1) be written Mn=(a + /3) M„_i- a;SM„.2, (a) a and ^ fulfilling the conditions a + ^ = 1 and o^ = -1 (6) From Equation (a) we have «n-OM„-l = /3(l(„_i- au„.-2) (c) This being true for all positive integral values of n, we may also write M„_l — OM„_2 = ;8(m„_2 - aUn-s), M„-2 — aU„-s = /3(m„_3 — aUn-i), Us — aU2 = P(U2 — aui). Hence Equation (c) may be written M„ — Ott„_i = (?«2 — aMi)/3»-2 ((J) Similarly, «„-/3tt„_i = (m2-|8!(i)o"-2 (e) Eliminating u„-i from Equations (d) and (e) we obtain _ (U2- aMi)/3«-' - («2 - ^Mi)a"-i , ., /3-a Equations (6) give the values of a and j3, and by inspection we have M2 = 2 and ui = 1. Equation (/) thus becomes U _ 1-V5 \ / l+V5 \"-i_ /g _ 1+ V5 \ / l-V5 \"-i M,i = — , V5 which is the same as Equation (2). 182 THEORY OP DETERMINANTS. [chap. vii. The value of m„ has been obtained by Professor Sylvester in the form ^ , , ^. , (n-2)(n-3) M„ = 1 + (n - 1) + -^^ ^^ ^ I («-3)(n-4)(n-5) 1-2.3 99. The determinant Q„ is equal to gn = «! X2 • • C2 tl2 ^3- . C3 as • . . •• Ctn-l \, • •• c„ a, (1) when — XjCj = b^. For, developing in reference to the elements of the last row, we have Qn = an^n-l " KCnQn~2, or Qn = anQn-l + Ks \( h h \ [1 Oi Oj aj-"/ Vai + &i Og 03---/ 192 THEOKY OF DETERMINANTS. [chap. vii. 3. ( ^2 h— K \ 4. 11 1... 1 \ / I... 1 aiX~^ a^ a^x'^ a^x--- fl„a5<~""/ \a, a2-" a, or = a; VI... 1 > according as n is even or odd. 5. Obtain the result in Article 99 by multiplying each row of the determinant (1) by , and each column by — A,^i ; i being, in each case, the number of the row or column multiplied. 6. Express the series 1 + 3 + 5 + 7+.. .+(2n-l) in the form of a continued fraction. 7. Express the harmonic series i+Ul+i+...+l 12 3 4 n in the form of a continued fraction. 8. Show that ( \ h- K &i &2 h K \f^ «! «2 ••• a, ai + a2 + a3+ + «„ f &2 h--- K a, a., a, •■• a. ART. 105.] CONTINUANTS. 9. Show that the periodic continued fraction 193 63 &2 ^i I b, bo 6, * «! + «2 + a^ + +a.i + a.+ IJL + fn. + * where the asterisks indicate the recurring period 01 repetend, is equal to bi b,--- 63 b., bi ■ a.2 tti fj. bi &2 ■ • • &3 ^2 ^1 tti tto a^ ■ '3 ^2 bi \ 10. Show that 1 I." 1 111- I/ .-. ^ 2 (1 + V5)"-(1-V5)" / 1 I--- 1 \ ''(l + V6)"+'-(l-V5)"+i' \i 1 1... i;„ where w, or n — 1, indicates the order of the continuant to which it is affixed. 194 THEORY OP DETERMINANTS. [chap. viii. CHAPTER VIII. JACOBIANS, HESSIANS, AND WEONSKIANS. In the applications of the calculus there fre- quently appear determinants whose elements are the differential coefficients of systems of functions. Most of the determinants originating in this man- ner come under one or another of the classes treated in the present chapter. JaeoMans. 106. Let ?/i, 2/21 •••) y» be n functions, each of n independent variables ; thus : y« = /«(«!, «2, •■-, a;,.), j If now a determinant |ai'"| be formed in which a,(.)=%i, thus: ART. 107.] JACOBIANS. Sv, ^■2 hn 6a;i Sxi Sxi 82/2 8a;2 " 8x, hi 8x„ 82/2 6a;„ ■ S2/„ 8a;„ 195 this determinant is called the Jacobian* of the functions ^/j, ygi "••> ?/n with respect to the variables a;j, aTj, •••, x^. It is commonly denoted by ^fi) . or J{yi,y2, •••, 2/J- Note. — If we have a single function of a single variable, as y =/(x), the corresponding Jacobian is the differential coefficient -^. This fact lies at the basis of the close analogy which dx we shall find to exist between Jacobians and ordinary differential coefficients, and suggests the first of the above notations. The second notation should be used only when there is no possible ambiguity in regard to the Independent variables. 107. If the functions i/^, y^, •••, y^ are linear with respect to Xj, x^, ••-, a;„, thus : Vi = aii^i + «2ia52 + ••• + a,,ix,„ 2/2 ^ ClllXi + a.2:;PC-2 + ••• + 0,n2^M Vn = ai„Xi + a^^x.^ -\ + a„„a;„ ; * These functions were first studied by the Jewish mathema- tician Karl Gustav Jacob Jacobi (see Crelle's Journal, 1841), after whom they were named by Professor Sylvester. 196 THEOKY OF DETERMINANTS. [chap. viii. then the corresponding Jacobian is the determinant of the coefficients of the linear functions, or Jiy\> 2/2) ••■) 2/n)= I «!! a22 •" «»„!• This follows directly from the definition of a Jacobian. 108. If the given functions y-^, y^, ■■■, y„ are independent, we may express x^, x^, •••, x^ as so many functions of y^, y^, •••,y»- In this case we have the following relation : d(yi,y2, ■■■,y,d d(xi,x2, ■■■,x„) d{Xi, x,, ..., x„) d{y^, 1/2, •■-, 2/„) = 1. This may be shown by writing out each of the above Jacobians in determinant form, changing columns into rows in the first, and multiplying by rows. Thus, for n = 2, we have %1 §h 8xi 8x2 8x1 8x2 82/1 8y, 8yi 8yi 8x1 8x2 8X1 8x2 82/2 82/2 8^] 8^1 8^ 8^2 8^1 Skj 8^ 8^2 8x1 81J1 8xn 82/1 8x1 82/2 8a;2 82/2 8^2 8«i , %2 8x2 82/2 8x1 82/2 8.'«2 8x1 82/1 8x0 82/1 8xj 82/2 8x0 83/2 AKT. 110.] JACOBIANS. 197 8^ Syi 82/1 82/2 = 1 1 8^2 8^2 8yi 8^2 = 1, since y^ and y^ are independent. 109. If ?/j, 2/21 •••! 2/» are functions, each of o-j, erg, ■ ••, o-„, and if these in turn are functions, each of *^i' ^^2' *"^ *^*ti Liien (^ (yi. ya ■ • ■, y,.) ^ d (y,, yg, • • ■, y„) _ c?(o-i, 0-2, •••, o-,.) ^ d (xi, a;2, • • •, a;„> d (rrj, a^, ■ ■ ■, cr„) d (x^, Xj, • • •, a;„) This may be demonstrated by the method em- ployed in the preceding article, remembering that dy^^^dy^ dcij_^dy^ da2_^ dyi_ da„ ^ dx^ c?Tj dxj dfT2 dxii da-„ dx,, 110. If t/j, 3/21 ■■■' y« are given onlj' as implicit functions of x-^, x^^ •■■, x„, thus: i(yi> ■■■,yn,Xi, •••, a;„) = 0, "^2(2/1. "vyn, *!, •■•, a;„) = 0, then n{yi, ■•■,yn,^i, •••, a'J=0; <^(<^l! 2> •••) "^n) d{yi,y2, ■•■,yn) _ . js„^_(^i^_a»_^-^_a^ d(a;i, a;2, •■•, a;„) d((^i, ;, ••■, ^„) ' <^(2/i, 2/2, ••■, 2/n) (1) (2) 198 THEORY OF DETERMINANTS. [chap. viii. Writing the above Jacobians in determinant form, changing rows to columns in the first member, clear- ing of fractions, and indicating the resulting product in the first member by P, gives P = 8^, 8y,_|_S^%2^....^ Syi ?ix, 82/2 K HiSy^l. . . . (3) Let us now form the total derivative of the function t^,- with respect to %. We have it 4>2} •••> "^n) d(»i, X2, •••, a;„) which proves the given theorem. ART. 111.] JACOBIANS. 199 111. The reader will doubtless have noticed the analogy between the formulae developed in the three preceding articles, and the following well- known formulae of the differential calculus : where y=f(x); dy dx _ ^ dx dy dy _ dy da dx da- dx wnere y=f(^cr), and cr = ^(a;); and d4 dy _ dx dx dcj} dy where (f)(y, a;) = 0. This analogy led Bertrand to devise a definition of a Jacobian which should itself be analogous to the definition of a simple differential coefficient. Bertrand's definition is as follows : Referring to Equations (1) of Article 106, let AiKi, ^iX,, ■■; AjX„, I AjK], A2X2, •••, A2a;„, I ,^, A„a;i, A„a;2, •••, A„a;„, J be n distinct series of increments given to the n variables x-^, x^, •••, x , and 200 THEORY OP DETEEMINANTS. [chap. viii. A22/1, A22/2, ■", A22/„, A„3/i, A„2/2, •••, A„2/„ (2) the corresponding increments assumed by the func- tions. Then, just as the limiting value of the ratio -^, that is -f-, is the differential coefficient of y Arc dx ^ with respect to x when y=f(x), so is the Jacobian of «/j, 2/2, •••, «/„, with respect to x^, x^^ •••, a;„ the limiting ratio of the determinant of the systems of increments (2) and (1). Thus, rt(,Vi, ya •••■y») d(a;i, a;2, ..., a;„) di?/i dji/i di2/2 d2?/2 dl2/» ^22/,. ••• rtn^n diXi c\o',2 ••■ diX„ d„Xi d„X2 ■■■ d„x^ This may be shown by clearing the above equation of fractions, and using the multiplication theorem, remembering that f^t2/,= ^ dja-i -f -^ dtX2-\ f-^ d^„. dXi da-j 6X„ ART. 112.] JACOBIANS. 201 This definition may be used as a basis for the proof of each of the preceding formulae. It is the natural basis of the arbitrary definition of a Jacobian first given. 112. The Jacobian of a set of n functions, y^, y^, •••, y„, each of the n independent variables Xj, Xj, •••, x^ is equal to the product Sxi 82/2 Sy„' in which yjff^ is a function of ji, •■■, y^..,, Xk, ■••, x„. Letting t/^ = -v^^, (ajj, x^, •■■, x ), we may obtain x^ as a function of the variables 2/v ^2^ ••■1 ^n- Substituting this value of x, in the equation giving 2/2, we have t/^ = yjr^ (^y^, x^, •■•, x„). This equation will make known x^ as a function of y^, y^, Xg, •••, a;„, which, substituted in the equation for y^ gives Vz = ■^s iVv Vv H-> ■■- ^n)- We thus obtain the system y\ = '/'l ('''I) *2) ^3) ■■■; ^n)> 2/2 ^ "As {Vu ^> %' ■ ■ ■' ^n)! Vk = "A* (2/1, 2/2, •■•, 2/*-i, «« •••, «»). 202 THEORY OF DETERMINANTS. [chap. viii. Forming the Jacobian of this system by the method of Article 110, remembering that and that for k < i, we have Hence 8i//,._ 8% = for A; > I, d(yi, V'l, ■■; y„) _ ^_i\n Cl(Xi, X2, •■■, x„) = (-1)" Hi Sxi ••• 8^1 8x, 8X2 •■• 8X3 -^-h ... 1 1 - S2/1 Sl//3 82/1 1 - 8i/'s 8^/2 1 ... djyi, ya ••-. yn) _ Mi . ^-...^ d{Xi, X2, ••; x„) Sxj 8x2 Sx„ 113. If the Jacohia7i of a set of functions vanishes, the functions are not independent. AET. 114.] JACOBIANS. 203 For, if a{^Xi, X.2, ••■, a;,,) we have by the preceding article in which k has some value between 1 and n. Then i/tj does not involve a;*, or Vk^-^kiyy, ■■■, 2/t-i, £»*+!, •••, «„)• Eliminating x^+i between this equation and gives a result which may be written in the form J/i+i = J'i+i (2/i> •••) Va ^k+2! •■•) 2;„) ; that is, 2/j+i does not involve 0:4^1. In the same manner it may be shown that 3/1^3 •iocs not involve Xk+2, •", and finally that «/„ does not involve x„. Then yn = F{y^,y2, ■■; y„-i), or the functions ^/j, y^, •••, y„ are not independent. 114. We shall now demonstrate the converse of the theorem in the last article ; that is : If the functions y^ j^, •••, 3'-„ are not independent, the Jaeohian vanishes. 204 THEORY OF DETEKMINANTS. [chap. viii. Let /(2/1, 2/2, •••, 2/,.)=0 be the equation connecting tlie given functions. Differentiating tliid function witli respect to each of the variables Xp x^, •■■, x„ gives the consistent system, V 8£. + ... +«/?£" = 0, 8^1 Sxi By,, 8xi S2/1 8a;„ 82/„ 8a;„ Eliminating -/-, •■•, —-from these equations gives 8^1 ... 8^„ Sail 8a;, Sj^i ... %. Sa;„ S-r,, = ^(yi, ^2, ■••- 2/..)= 0- 115. If, in Article 109, the functions o-j, a^, •••■, 0-,, are linear, thus : <7, = a„a;i + a^ajj H f- a,aX„, 0-2 =: ajjaji + 0220:2 H l-a„2a;n, (1) ART. 116.] HESSIANS. 205 then d(a;i, Xa, •••, «„) d(cri, o-j, •••, o-„) ' in which /i is the determinant This follows at once from Articles 109 and 107. We have here a general case of the linear trans- formation of a set of functions, a special case of which has already been noticed in Article 54. The determinant fi is, as before, called the modulus of transformation. The contents of this article may now be summed iip as follows : If^ a set of functions he subjected to linear trans- formation, the Jacohian of the transformed functions is equal to the Jacohian of the original functions multiplied hy the modulus of transformation. For this reason the Jacobian is a covariant of the set of functions from which it is derived. (See Art. 128.) ffessians. 116. The Jacobian of the first differential coeffi- cients of a function of n variables, taken with re- spect to the several vai'iables, is called the Hessian of the function. 206 THEORY OP DETERMINANTS. [chap.viii. Thus, if then is w=/(e n functions of a single variable x. If now a determinant ja^'"! be formed, in which thus: * -dx'->' y\ Vi -Vn dyi dy^ dx dx • dx d^-'yi d»- Ih. d"-'y,. dx"-' da;" -1 dx"-^ this determinant is called the Wronskian of the given functions with respect to x.* We have the following abridged notations : Wx(2/i,^2> ■••,2/»); 2/1 2// 2/2 2/2' •• 2/» ••2/»' 2/1'"-" .%'»-» -I/,/"-'' (1) * This name was given by Mr. Muir, after the Polish mathe- matician Hoene' Wronski (1778-1853), by whom the function was introduced in connection with the expansion-theorem which also bears his name. The "Wronskian had been called simply the deter- minant of the functions, and written I>{yi, y^, ••■, Vn)- AKT. 121.] WEONSKIANS. 211 in the second of which the order of the derivatives is indicated by superscripts. 121. Letting a- be any function of x, we may write arbitrarily, o-" = 0- 2(t' 3o-" ••• f-V''-""Y^^V<"-'> 1 n- 3(7' ... /'iy<'-=>.-Y'i^y("-=» o- ... fiV'"-'' ••/-"- V"-*' 3/ V 3 , (2) ... ••• a- in which 1, f M, r-A, (M, ••■, 1 are the coefficients of the expansion of (a + 6)'. Multiplying together Equations (1) and (2), the second members by columns, we have yifc,.,..,,., = w[g)',(|J,....(?.)],(l) the second member being of the order (n — V). But (y^')=yiynml=\w{y.,y,). Letting TF"(s'i, Vi) = ^ii Equation (1) becomes, by means of the last article, W(yi, 2/2, • • •, 2/„) = -^ W{W2, W3, ••.,«;„) . . . (2) 2/1 123. Jf the functions are connected by a linear relation, as ai2/i + a22/2 H h anVn = 0, the Wronskian vanishes. ART. 124.] WRONSKIANS. 213 This may be shown by eliminating aj, a^, •••, a„ from the above equation and its first (n— 1) deriv- atives. 124. The converse of the preceding theorem is also true. That is : If the Wronskian of a set of functions vanishes, the functions are connected hy a linear relation with constant coefficients. We shall employ the method of induction. If, then, W{yi,y2, •■■,y„) = 0, we have, by Article 122, Tr(t«2, Wj, .", w„) = 0. The theorem being assumed true for (n— 1) func- tions, we have ajW2 + ajTOg -) 1- a„io„ = 0; or, dividing through by y^^ This gives in which a^ is the constant of integration. 214 THEORY or DETERMINANTS. [chap. vm. Hence, if the theorem is true for (n — 1) functions, it is also true for n functions. But, as is readily seen by inspection, it is true for two functions, and therefore generally. EXAMPLES. 1. JFind the Jacobian of the functions 2/i = 3:2" + 2 aix^x^ + xi, 2^jj ^ aji -\- z o, •••) 2/n) _„, 7.-1 ™n-2 ™2 ~ — — — U,i 0/2 •■■ •<'n-2 •''m-1- CH^Xi, X2, •••, X„) 3. If the functions are as follows: 2/1 =fi{^i), 2/2 = /zV^'lI *2)) ■"> 2/»^^yn(*'l) '*'2l ■■■? •">!)) Show that j=8yi.SL2...%„. 8x1 Sa;2 8x„ ART. 124.] JACOBIANS, HESSIANS, WRONSKIANS. 215 4. Find the Jacobian of y^ 2/2, •••, y„, being given 2/i = (l-a!i), ys = x^x.,{l-Xi), ■■; y„ = XiXi---x„_^{l-x„). Ans. {-lYxr'x,''-'...a?„_,x,_,. 5. Find the Jacobian of a-'i, X2, •••, x„ with respect to ^i> 62, ■■■> 0^ being given Xi = cos ^1, x^ = sin ^1 cos 0-2, Xg = sinOi sinOi cosO^, ••■, a;„= sin^i sin 62 ••• sin6„_i cos^„. 6. Given m = (r + a cos^)^, v = (r — b sin 6)^, in which r = (x" + f )^, 6 = arc tan ^ . to find ^Ii}hJl . x d{x,y) 7. Given yi{xi — X2) = 0, y2{x{' + x,X2 + X2'') = 0, to find ^iy^^. Ans. 3m2-;^^V 8. Given ^-22ii* = 0, ^A^' ^p, to find ^^"' ^^ • V cos y y sin v d{x, y) . uv cos It siu-i; sin (a; + y) xy sin a; cosy sin (m + v) 9. If 2/j^i, •••, ?/„ are independent of Xi, ■■•, x^, or if 2/1) •••) 2/t s-re independent of ^t.^, •••, a;„, show that c^yi. •■•, y*. y^+i. ••-, .v, .) ^ ^kvi, •••. y*) . <^(.%+i, ••■, y„) dO-K], •••, %, a;t+i, •••, a;„) d{x^, ■••, a;^) d(a,Vi, ..., a-,,)' 216 THEORY OF DETERMINANTS. [chap. viii. 10. The conditions of the preceding example still holding good, show that djy^, •••. fe a-'t+i. •••) ^n) _ d(yi, ya ••-, y^) 11. If the functions yj, y^, ..., ?/„ are independent, show, by means of the preceding example and Article 109, that dJVi, ••■, y,.) _ d(a:„ ...,x,) ^ d{y^^.i, ■■■,y„) _ dixi, ..., a;„) ■ d {y^, ..., y,) d{x^^i, ..., x„)' 12. Prove the formula of Article 109 by means of Bertrand's definition of a Jacobian. 13. Find the Jacobian of the functions, 2/i= (Xi — x.;)(x, + Xs), 2/2= (a;, + a;,,)(xo-a;3), 2/3 = ^'2(^1 — ^s)) and thus show that the functions are not independent. 14. Find the Jacobian of u = x{y -\- z), v = y{x-\-z), w = z{x — y), and thus show that u, v, w are not inde- pendent. 15. If a; = r cos^, y = r sinO, in which r = mi, 6 = hv, find ^(^^. , , d(u,v) -Ans. abr. 16. Given, u = x + y, v = xy, m. which X = x' cos a — y' sin a, y-=x' sin « + y' cos a, to find ^(^^'^) . d (a, y) AKT. 124.] JACOBIANS, HESSIANS, WRONSKIANS. 217 17. Find the Hessian of the function, aa;- + hy" + c£- + 2fyz + 2 gxz + 2 lixy — 0. 18. Find the Hessian of the function given in the last example upon introducing the substitutions, X = l-^x' + m,?/' + Til?', y = l^x' + mjj' + 11^', z = l^x' + m.oy' + Wjg'. 19. Find the Hessian of the function, Ans. 36 ago a2i 0^1^+ a3o 0,12 ttji cin a;ia;2+ ftoi (Xi2 *21 "03 CI12 ^03 a;,^|. 20. Show that, if a homogeneous relation exists among the functions u, Vq, ..., ?t„, then is 21. If Wv = ~\ show that t K (U, Ml, . . ., M„) = — K{V, V,,..., v„) . 22. Prove that, if y„ y^, ■■■lyn are functions of o-, and o- a function of x, then is TF.(yi,y2,...,2/„) = (^|^^ ' W,{y,,y„ ...,y:). 23. Show that the total differential of a Wronskian with respect to its independent variable is obtained by differentiating the elements of the last row. 218 THEORY OF DETEKMINANTS. [chap. vin. 24. Prove the formula, W\W(yi,y2,y3), W{yi, y2,yi),...,W{yi>y2,yn)\- [_W{y,,y.^-T-' 25. Generalize the preceding example. 26. Find the Wronskiau of the functions cc" + a, x" -\-b, x" + c, and thus show that these functions are connected by a linear relation. Also, find this linear relation. 27. Show that a linear relation exists among the functions, sin x, e^'^'\ cos x. ART. 125.] LINEAR TRANSFORMATIONS. 219 CHAPTER IX. LINEAR TRANSFORMATIONS. The tlieoi-y of linear transformations, in its appli- cation to quantics, and particularly to quadrics, is of the greatest importance in connection with the study of modern geometry. The student wishing an extended course in this subject should read Salmon's Modern Higher Algebra, or perhaps better still, Clebsch's Vorlesungen iiber Geometrie, where it is presented in its true geomet- rical relations. The greater number of the articles in the present chapter are capable of direct geometrical interpre- tation ; though that interpretation is only hinted at in two or three instances, being outside the scope of the present work. 125. A quantic is a homogeneous function of any number of variables and of any degree. The number of variables is indicated by some one of the adjectives : binaiy, ternary, quaternary, etc. ; while the function is called a quadric, a cubic, a quartic, etc., according to its degree. In 220 THEORY OF DETEKMINANTS. [chap. ix. general, the quantic involving n variables and of the mth degree, is called an wary mic. Thus, as an example of a binary cubic, we have a3oa;i'' + 3 a^Xi^ + 3 a^aiX-f + Uo^x^. This is commonly denoted by the symbol when the numerical coefficients are those of the expansion of (2:^ + 2:2)^- When the terms are not affected by numerical coefficients, the notation is ("■SO) '^21) "^izj ''osx"'!' ■''2) — '^aO'^'i 4"Cl2i^r'*'2 + C(i2a;iX2 + ctmXj'. The same notation is used for quantics in general. 126. The discriminant of a quantic is the elimi- nant of its first derivatives taken with respect to its several variables. Thus, if g = ci^xi ~\- o cijiX'i ^2 + o ci^^iX^ -f- Oq^x^ , we have Q\ = ^^ O CtgyX]^ —J— U Ct2i*^l*^2 "i '-^ Otj2*^2 j Sxi q2 = -^ = o ctiiXi^ + 6 a^iXyXi + 3 a^^xi' 8x. and the discriminant is the eliminant of q^ and q^, which is (Art. 47) ART. 128.] LINEAR TRANSFORMATIONS. 221 3a3o 6a2i Sajj Sctjo 6a2i 3a,2 3 a,! 6 a, 2 Sug^ Scioi Gttj, 3ao3 ^ 127. An invariant is a function of the coefficients of a quantic which is not affected by linear trans- formation of the quantic, excepting that it is multi- plied by a power of the modulus of transformation. Thus, if the quantic q in the preceding article be transformed to q' by the linear substitutions ajj = buUi + l>i2U2, X.2 = &21M1 + &22*f2) and if the discriminant S' of the quantic q' be formed, we shall have 8 = S. &„ &,2 Hence 8 is an invariant of q. 128. A covariant of a quantic is a function involv- ing both the coefficients and the variables of the quantic and so related to it that, when the quantic is subjected to linear transformation, the same func- tion of the new coefficients and variables shall equal the original function multiplied by a power of the modulus of transformation. We have already noted the Hessian as a covariant of the function from which it is derived. (Art. 117.) 222 THEOKY OP DETERMINANTS. [chap. ix. As an example of a covariant of a set of functions we have the Jacobian. (Art. 115.) It follows at once from the definitions that every invariant of a covariant is an invariant of the original quantic. 129. The quadric in n variables, a;j, x^, •••, a;„, is usually denoted by q = •S.taaXiX^, in which the coefficient of a:/ is «„■ and that of xfc^ is 2 da, a,i leing identical with a^i. We have 5'i = 2 g^ = «ii*"i + «i2 a^s H h a-m^n, \ ^ — /Vj Jjii ^ JD22 =^ 1> and ART. 134.] LINEAR TKANSFORMATIONS. l-X^ 2X ■2X ■ Y^„ ay, - ^^^, a,, - ^-^^, a,, = 1 231 1 + X." which gives for the required linear substitutions : 2A. 1-X2 X, = u. X2 — - •2X 1 + X' Ml +; 1 + A^ ' ' 1 + A^ '' in which X may be chosen arbitrarily. For the ternary orthogonal tianst'ormation we have, X, fi, and V being arbitrary, B = 1 V -V 1 fJL X SO that B„ = 1 + X^ i?,, = V + X/i, B,, = ,ji + \v, -B21 = — 1/ + X/i, -Bi2 = 1 + F^ B.^ = X + fiX, •^31= M+'^'') -^32=— X + /XV, ^33=1+^1 Hence the coeiBcients are, in order, 1 + X^-j^^-i 2 — V + X/ i. 5 ' , 2 -/<. + Xv 5 ' ^~s~' Q — X + /XI/ 1 _ X^ _ ^2 ^ ^2 5 The coefficients for orthogonal transformations of higher orders may be found in the same manner. 232 THEORY OP DETERMINANTS. [chap. ix. 135. Variables which are subjected to the same linear transformation are said to be cogredient. Thus, if a function of x^, ..., x is transformed by the substitutions Xi =auUi -\ |-«i„M« and a function of x\, ..., x'^ by a;'„ = a„]M'iH h «„,y,„ then x-^, ..., x„ are cogredient with x\, ..., x'„.- Note. — When, in analytical geometry, we transform to new axes, the co-ordinates (x, y, «), (x', y', «'), •••, expressing different points, are cogredient. Let us transform the function ^ (a;^, x^) by means of the substitutions ajj = ajj + ja; 1, X2 = X2 -tJX 2, where x-^, x^ are cogredient with x\, x\. Developing the new function thus obtained b}"- means of Taylor's theorem, we have ART. 136.] LINEAR TEANSFORMATIONS. 233 i.{x,+ jx\, X, + jx\) =.^+j (x\^ + x\^ + -^—?,J'[ ^ I —^„ + Sk'is; 2 — ^— + a; 2 — ^ 1 • ^ \ Sx-f 8a;iSa;2 hx^, + ■••• In the above development, the coefficients of j, ——fi ..., — ri" are called the first, second, ..., w"' 1- z «! emanents of <^ (a;j, aij). The wth emanent may be symbolically represented by 8a;] 6x^J It may readily be shown that these emanents are covariants of the function ^(xi, x^. The reader will find no difficulty in extending the above to any number of variables. 136. Let us assume two sets each of n variables having the relation U^X^ + U2X2+ ••• +U„X^ = Q. (1) If this quantic be transformed by means of the linear substitutions aji = aiia;'i + a^x'^ H h «!„»'„, ' X^ = d^^ 1 -\- Ct2zX 2 ~r ' * ■ ~r ^hn'^ nJ Xj^ — ^^nl*^ 1 *r ^^712*^ 2 ~r ' ' * T" ^nn*^ nJ (2) 234 THEORY OF DETERMINANTS. [chap. IX. it will become H h («inMi + a2„M2 H 1- «»««») «'» = 0. . . (3) Writing an Ux + ^21 Mj + • • • + «niW„ = m'i > HiaWi + 022^2 + ••• + Cfn^^'n = W'2, (4) J a,„?.ti + a2„M2 ^ h «„,.'<» = che function (3) becomes u\x\-\-v.\x'„^ 1- «'„«'„ = 0, (5) which is of the same form as the original quantic. Solving Equations (4) for Mj, Mgi •••■, u„, represents ing the determinant of the system by S and the co-factor of a.j by .4,4, gives 8-Mi = Aii2c\ + Ai2u'-2-] + A,„u'„, -j 8 • M2 = ^21 w'l + ^22 M'2 + h -^a-iM'.. 8 • M„= ^„iM'i + ^„2M'2 -I 1- A„„u', (6) If, then, the quantic (1) be subjected to a linear substitution which leaves its form unaltered, the elements of the modulus of substitution for one set of variables, Mj, u^ ..., m„, are the co-factors of the corresponding elements of the modulus for the other set, X^, 3^2, ..., ^}f ART. 137.] LIKEAK TRANSFORMATIONS. 235 Two sets of variables related in this manner are said to be contragredient or reciprocal. Note. — The geometrical meaning of the above is that, in changing to a new system of reference, the triangular co-ordinates of a line are contragredient to the trilinear co-ordinates of a point. In Article 129, the semi-differential coefficients q^, q^, ..., q^ are contragredient to x^, x^, ..., x„, it being remembered that a^|. = a^^. 137. Resnming the equation UiXi + v,^2-\ t-Mna;„ = 0, (1) let «„ = a«i^i + <^«'J«2 -\ 1- a-nn^n, where a^ = an- (2) Substituting these values of Mj, ..., w„ in (1), gives ttiia;,^ -1- a.^.f -\ \-2 ai-M^x.^ + 2 a^x^x^ -\ -t- 2 a2i.x^s -\ = 22a,4a;,a;t = q. Now, Equations (2) show that x^, ..., x are contra- gredient to tip ..., u„. Hence, substituting in (1) the values of Xy ■■■,x„ expressed in terms of Mj, ..., u„ gives AuUi' + A<.iui -\ 1-2 A^M^ih + 2 A i^u^iis -\ + 2A^nM.i + ••■ s 22^««,Mt =Q. 236 THEORY OF DETERMINANTS. [chap. IX. The quadric Q is said to be reciprocal to q. It may be written in the form (Art. 38) Q = - «11- •«I„ Ml «„1- •Oto U„ Ui ■ •M» a form introduced by Hesse. Note. — Geometrically, if the two quadrics are ternary, q = is the trilinear, and § = the triangular equation of a conic. Equation (1) represents a tangent to 5=0, or a point on § = 0. If we also have Ss As ttn •■•«,„ = 0, Ai--- A„ -4„i • • ■ -4„, (Art. 60) ; that is, if q is composite, so also is Q. But further, if S = 0, we have, by applying Article 62 to the symmetrical determinant S, /( 2 A 4 . hence, using an obvious notation, = (2aM)'; ART. 137.] LINEAR TRANSFOKMATIONS. 237 that is, If the discriminant of a quadric vanishes, the recip- rocal quadric is a perfect square. (See Ex. 24, after Art. 50.) EXAMPLES. 1. Write the discriminant and tlie Hessian of the ter- nary cubic. 2. Find the values of Tc for which the quadric (ix' + 3z" - 5?/2 + 3xz + '2xy) + k{x^ + 3y'^-yz + 5xz + 3xy) = is resolvable into linear factors; that is, represents a pair of right lines. Write, if possible, the square root of the reciprocal of each of the following quadrics : 3. 3x^-6y--5z- + 13yz-14:xz + 7xy = 0. 4. x- + 3xy — xz — 0. 5. wx — wz — x''-\-xy-\-xz—yz=0. 6. a-x'-2abxy + by = (). 7. xP — y^ + z^ — iyz + 6xz — 2xy = 0. 8. 2xy-y^--6xz + 3yz = 0. 9. Show that the substitutions x = x' cos a — y' sin a, y = x' sin a + y' cos a, are orthogonal. 238 THEORY OF DETERMINANTS. [chap. ii. 10. Show that the substitutions a; = a;' cos «i + y' cosySj + 2' cos yi, 2/ = x' cos a^ + y' cos fi^ + «' cos y.^, Z=x' cos «3 + ?/' cos /Sj + 2' cos yj, are orthogonal; («i, /Ji, y,), (aj, /?2> 72); («3) A, 73) l>eing direction angles in Cartesian co-ordinates. 11. Write the coefficients for the quatenary orthogo- nal transformation. 12. Prove, by means of Equations (3), or (5), of Article 133, that any element of the modulus of an orthogonal substitution is equal to plus or minus its co-factor. 13. Prove that any minor of the modulus of an orthogonal substitution is equal to plus or minus its complementary co-factor. 14. Show that the emanents of a quantic are all of them, in general, of tlie same degree as the original quantic. 15. If the quadric («, b, c,f, g, hjx, y, 2) be transformed by the linear substitutions x=-x'+ y'+ 2', y= x'-2y'+ z', 2= x'+ 2/' -32', write the corresponding substitutions for the reciprocal quadric {A, B, C, F, G, HJu, V, w). INTRODUCTORY MODERN GEOMETRY OP THE POINT, RAY, AND CIRCLE. WILLIAJiI B. SMITH, Pli.D., Professor of Mathematics in Missouri State University. Price, $1.10. This book is written primarily for students preparing for ad- mission to tlie freshman class of the Missouri State University, and has already been thoroughly tested in the sub-freshman department of that institution. It covers both in amount and quality the geometrical instruction required for admission to any of the higher universities. From Prof. George Bruce Halsted, Pli.D. (Johns Hopkins), Professor of Mathematics, University of Texas. "To the many of my fellow-teachers in America who have questioned me in regard to the Non-EucUdean Geometrj- 1 would now wish to say pub- licly that Dr. Smith'.s conception of that profound advance in pure science is entirely sound. . . . Dr. Smith has given us a book of which our country can be proud. I think it the duty of every teacher of geometry to examine it carefully." From PrincipalJoHN M. Colaw, A.M., Monterey, Va. "I cannot see any cogent reason for not introducing the methods of Modem Geometry in textbooks intended for first years of a college course. How useful and instructive these methods are, is clearly bi ought to view in Dr. Smith's admirable treatise. This treatise is in the right direction and is one step in advancing a doctrine which is destined to reconstruct in great measure the whole edifice of Geometry. 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