^^(/ 
 
 ^9^ 
 
 ^S 
 
 CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 GIFT OF 
 
 Department of Math 
 
Cornell University Library 
 
 arV19323 
 
 Determinants. 
 
 3 1924 031 245 826 
 olin.anx 
 
Cornell University 
 Library 
 
 The original of tliis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031245826 
 
PREFACE. 
 
 Sylvester said, "Determinants is an algebra upon an 
 algebra; a calculus which enables us to combine and fore- 
 tell the results of algebraical operations in the same way 
 as algebra itself enables us to dispense with the perform- 
 ance of the special operations of arithmetic." 
 
 A knowledge of determinants is essential to the reader 
 of modern mathematical periodicals and to the student of 
 the modern higher mathematics. This knowledge should 
 be attained by the student early in his mathematical studies. 
 
 The present work is designed to simplify the methods 
 of determinants and furnish a text suited to the needs of 
 students possessed only of a knowledge of elementary al- 
 gebra. 
 
 It was written and tested in the class-room as an intro- 
 ductory course to coordinate geometry. The test was 
 satisfactory, and believing that the book might be useful 
 to students of high schools and the lower classes of col- 
 leges and universities, the author offers it to the mathe- 
 matical public. 
 
 In preparing the work gleanings have been made in the 
 mathematical fields far and wide. However, the plan of 
 the book belongs to the author, but he is much indebted 
 to many authors and mathematical periodicals. Among 
 them it is due that he acknowledge the writings of, Hanus, 
 Weld, Peck, Bumside and Panton, The Analyst, and The 
 
i PREFACE. 
 
 Annals of Mathematics. To Burnside and Panton acknowl- 
 edgments are due for many exercises in this book, and in 
 a less degree to the other works mentioned. 
 
 It is believed the book will find a place not occupied 
 by any other of its kind. 
 
 There seems to be some confusion among writers on 
 determinants in regard to the technical terms ' ' element ' ' 
 and "constituent" and believing that mathematical ter- 
 minology should be precise and uniform among writers, the 
 author of this work has suggested and carried into effect 
 a precise meaning of these terms ^ which innovation, if such 
 it may be called, will avoid confusion in the application 
 of these terms. 
 
 With the hope that his effort may help to popularize 
 this young and important branch of mathematics, the author 
 submits his work with some diffidence to teachers and 
 students. 
 
 Thanks are hereby extended to Thos. R. Vickroy, Ph.D., 
 of the Werner Company, for timely suggestions, and to E. 
 A. Crueger, student in the University of Washington, who 
 has kindly read the proof. 
 
 Criticisms of the book will be thankfully received. 
 
 J. M. TAYLOR. 
 University of Washingtoji. 
 
CONTENTS. 
 
 ARTICLE. PA(il-;. 
 
 1. Definition op a Determinant 7 
 
 2. Definition op an Ei<ement 7 
 
 3. Definition of a Row 7 
 
 4. Definition of a Coi^dmn 7 
 
 5. Orders 7 
 
 6. Constituents 7 
 
 7. Principai, and Secondary Diagonals 7 
 
 8. Principal Diagonal 8 
 
 9 Secondary Diagonal 8-9 
 
 10. General Form of a Determinant of the Third 
 
 Order 10 
 
 11. Determinants op Any Order 12 
 
 12. I/Aw OF Formation of Determinants 12 
 
 13. Elements of Constituents 12 
 
 14. The Sign Symbol OP THE Constituents 12 
 
 15. The Symbol for the Number of Con.stituents .... 13 
 
 lG-20. General Properties op Determinants 14-15 
 
 21-22. Minors..". 16 
 
 23. Complementary Minors 16 
 
 24. Principal Minor 16 
 
 25. Co-Factors 16 
 
 26. DETERMINANTS OP THE wth Order 16 
 
 27-31. Additional Properties of Determinants 17-24 
 
 32. Solution op Equations of Two Variables 25-27 
 
 33. Solution op Equations of Three Variables 27-29 
 
 34. Solution of Equations of n Variables 29-31 
 
 35. Consistent Equations 32 
 
b CONTENTS. 
 
 36. EtiMiNANTS 32-33 
 
 37. Homogeneous Equations 34 
 
 38-39. Ratios of Variables of Homogeneous Equations..3-1-35 
 
 40-41. Product op Two Determinants 37-40 
 
 42. Reciprocai< Determinants 41-42 
 
 43. Conjugate Elements 43 
 
 44. Symmetrical Determinants 43 
 
 45. Axis of Symmetry 43 
 
 46. Zero-Axial 43 
 
 47. Skew-Symmetric Determinants 44 
 
 48. Zero-Axial Skew-Symmetric Determinants 44 
 
 49. Skew Determinants 44 
 
 50-51. Properties of Skew Determinants 44-48 
 
Determinants 
 
 DESIGNED FOR 
 
 High Schools and the Lower Classes of 
 Colleges and Universities 
 
 BY 
 
 ' J. M. TAYLOR, M.S. 
 
 Professor of Mathematics and Astronomy in the University of Washington. 
 AND Director of the Observatory 
 
 CHICAGO NEW YORK 
 
 Werner School Book Company 
 

 ■^y^rft 
 
 #f 
 
 er 
 
 -n , , 
 
 ^^^ ^<^^ 
 
 Copyright, 1896, by J. M. TAYLOR. M.S. 
 
 ^^^ of h\o-'W 
 
DETERMINANTS. 
 
 I. 
 
 PRELIMINARY DEFINITIONS. 
 
 1. A Determinant consists of w^ numbers arranged in 
 a square between two vertical lines. 
 
 In the equation r* ,/ = a^b^- a^ b. the first 
 
 member is called a determinant and the second member is 
 called its development. 
 
 2. Kach number or symbol in a determinant is called an 
 element. 
 
 When an element is a polynomial it is called a compound 
 element. 
 
 3. The elements in a horizontal line are called a row. 
 The rows are numbered y?r5^, second, third, etc. , beginning 
 
 at the top. 
 
 4. The elements in a vertical line are called a column. 
 The columns are numbered 7?r5/, second, third, etc., begin- 
 ning on the left. 
 
 Note. — In every determinant there are as many rows as 
 columns; hence, the. number of elements in a determinant is 
 a square number represented by n^ in which n represents the 
 number of rows and the number of columns. 
 
 5. If M equals 2, the determinant is of the second order; 
 if n equals 3, the determinant is of the third order, etc. 
 
 6. Each of the products of n elements in the development 
 of a determinant is called a constituent. 
 
 7. Every determinant has two diagonals called respect- 
 ively, PRINCIPAL and secondary. 
 
 7 
 
O DETERMINANTS. 
 
 8. _The elements standing in a line from the upper left- 
 hand corner to the lower right-hand corner constitute the 
 
 PRINCIPAI< DIAGONAL. 
 
 9. The elements standing in a line from the lower left- 
 hand corner to the upper ■ right-hand corner constitute the I 
 
 SECONDARY DIAGONAL. 
 
 To illustrate these definitions, we will take the third order 
 determinants 
 
 1. 2. 
 
 
 and 
 
 m + 71 a b 
 
 c r + s d 
 
 e f p-\-q 
 
 The development of the first is 
 
 The development of the second is 
 (m+7i) (r+s) {p+q) -\- ade -I- bcf — eb(r+s) —fd(in+7i) — ac{p+q) 
 
 The method of developing these determinants will be ex- 
 plained in a succeeding chapter. 
 
 «j, a^, b^, b^, f J , ^2, etc., are elements. 
 
 m 4- w is a compound element. 
 
 «,^ b^, c^ and m + n, a, b constitute t\\Q: Jirst row in these 
 determinants, respectively. a^^a2,a^ and m + n, c, e con- 
 stitute the first column in these determinants, respectively. 
 
 a^b^c^, (m + ?i) (r+ s) (p + q), etc., in the development of 
 the determinants, respectively, are called constituents, be- 
 cause every determinant is formed from some poljmomial so 
 con.stituted. 
 
 The elements a 1, /'o.Cj constitute 'Ocv^ principal diagonal oi 
 the first determinant. 
 
 The elements a.^,b^,c^ constitute the secondary diag07ial oi 
 the first determinant. 
 
 The elements m + ii, r + s,p + q constitute the principal 
 diagonal of the second determinant. 
 
 The elements e, r + s, b constitute the secondary diagoital 
 of the second determinant. 
 
DETERMINANTS. 
 
 • II. 
 
 DEVELOPMENT OF DETERMINANTS OF THE SEC= 
 OND ORDER. 
 
 RULE. 
 
 Subtract the product of the elements in the secondary 
 diagonal from the product of the elements in the principal 
 diagonal. 
 
 EXERCISE I. 
 
 Develop the following determinants: 
 
 1. 
 
 16 
 
 I2 
 
 31 
 
 2. 
 
 1-2 31 
 1-2 5I 
 
 3. 
 
 1-4 
 I 3 
 
 4. 
 
 10-31 
 
 2 
 
 5. 
 
 2I 
 
 7.* 
 Isin 9 sin * I 
 |cos 9 cos * I 
 
 10. 
 l^ogx logjl 
 
 6 
 
 13. 
 
 \x + y X — y\ 
 \x — y X + y\ 
 
 11 
 
 IcOS 9 
 
 \a + b 
 \a — b 
 
 cos 9 
 1 
 
 COS 
 — sin 
 
 sm 
 cos 
 
 16. 
 
 I 28 
 
 19. 
 
 \na y\ 
 \nb x\ 
 
 11. 
 
 a^ + ab + b^ 
 a^ — ab + b^ 
 
 14. 
 
 I — n —2 in\ 
 
 - 2 « n— m\ 
 
 17. 
 
 - 8 11 
 12 71 
 
 20. 
 
 12. 
 \x — y 
 
 \x^ _j/2 
 
 15. 
 
 hi" 1 I 
 ll n^l 
 
 18. 
 
 8 
 
 32 
 
 ll 
 
 l| 
 
 21. 
 
 la + mc 
 b 
 
 + mx\ 
 ax I 
 
 X 
 
 1 
 
 y 
 
 y 
 
 *The 7th, 8tli and 9th exercises may be omitted by students that 
 have not studied trigonometry. 
 
10 
 
 DETERMINANTS. 
 
 III. 
 
 DETERMINANTS OF THE THIRD ORDER. 
 
 10. The general form of a determinant of the third order 
 
 IS 
 
 b. 
 
 This may be developed as follows: 
 Repeat the first two columns on the right of the third; thus, 
 
 a. 
 
 b. 
 b„ 
 
 a „ 
 a. 
 
 b, 
 b. 
 
 + + 
 
 Then the development is a^b^c^ + b^c^a^ + c^a^b^ - 
 a^b^c^ — b^c^o-x ~ c^a^b-^. 
 
 This illustration leads to the following 
 RULE. 
 
 Form the products of the elements in the principal diagonal, 
 and in lines parallel to it, and mark them +. Then form 
 the products of the elements in the secondary diagonal, and 
 in lines parallel to it, and mark them — . 
 
 Note. — This rule will develop any determinant of the 
 third order. A little practice will enable the student to 
 develop the determinant by imagining the columns repeated 
 without writing them. 
 
 EXERCISE II. 
 Develop the following third order determinants: 
 
 1. 
 
 1 
 
 
 
 
 J' 3 
 
 y^ 
 
 2.* 
 
 
 
 
 3. 
 
 
 
 * 
 
 
 1 
 
 9 
 
 cos sin 
 
 * 
 
 
 .) 
 
 3 
 
 sm « cos 
 
 * 
 
 
 y 
 
 4 
 
 * Students unfamiliar with trigonometry should omit exorcise 2. 
 
DETERMINANTS. 
 
 11 
 
 
 
 
 4 
 
 
 
 
 
 5. 
 
 
 
 
 
 6 
 
 
 
 
 
 
 9 
 
 8 7 
 
 
 «1 «2 «3 
 
 
 X y 
 
 z 
 
 
 
 
 6 
 
 5 4 
 
 
 ^ ^2 ^3 
 
 
 Z X 
 
 y 
 
 
 
 
 3 
 
 2 1 
 
 
 fl f„ ^3 
 
 
 V z 
 
 X 
 
 
 
 7* 
 
 8. 
 
 
 9. 
 
 
 
 ^1 Ji 
 
 1 
 
 
 1 1 1 
 
 
 m + n 
 
 a 
 
 a 
 
 
 
 ^2 J2 
 
 1 
 
 
 X y z 
 
 
 b 
 
 n + a 
 
 b 
 
 
 
 ^3 Ji 
 
 1 
 
 
 x"" y^ z^ 
 
 
 n 
 
 n 
 
 a + b 
 
 
 
 10. 
 
 ll.t 
 
 
 
 12. 
 
 
 A c 
 
 b 
 
 
 1 CO,S X 
 
 cos y 
 
 
 
 1 2 5 
 
 
 
 c B 
 
 a 
 
 
 CO.S X 1 
 
 cos ^^ 
 
 
 
 2 4 2 
 
 
 
 b a 
 
 C 
 
 
 COS _y cos z 
 
 1 
 
 
 
 5 3 3 
 
 
 
 
 13. 14. 
 
 
 15. 
 
 
 
 1 
 
 
 
 
 4 1 S 
 
 
 4 1 
 
 5 
 
 
 
 
 3 
 
 «i b^ 
 
 
 6 2 6 
 
 
 2 3 
 
 
 
 
 
 
 3 
 
 «2 b% 
 
 
 8 3 4 
 
 
 6 2 
 
 1 
 
 
 
 
 16. " 17. 
 
 
 18. 
 
 
 
 
 a 
 
 b c 
 
 
 i i i 
 
 
 
 A 
 
 .ff 
 
 D 
 
 
 
 
 
 
 d e 
 
 
 i i \ 
 
 
 
 D 
 
 C 
 
 E 
 
 
 
 
 
 
 / 
 
 
 \ i ^ 
 
 
 
 D 
 
 E 
 
 F 
 
 
 
 19. 20. 
 
 
 
 21. 
 
 1 
 
 a^ + ^2 a-^a 
 
 2 
 
 1 
 
 5i 
 
 1 
 
 5 
 
 1 
 
 b^ + b^ b^b 
 
 2 
 
 -6 5 
 
 -17 
 
 
 -6 
 
 5-17 
 
 1 
 
 c^ + 
 
 ^2 fi C 
 
 
 7-17 
 
 6 
 
 
 1 
 
 6 
 
 
 
 22. 23. 
 
 
 
 24. 
 
 
 5 
 
 76 
 
 
 (5f « 
 
 flg 
 
 
 
 
 a, b^ 
 
 
 
 -17 
 
 76 -103 
 
 
 ra d 
 
 ^2 
 
 
 <^i 
 
 b^ 
 
 
 
 6 - 
 
 -103 
 
 
 ab c 
 
 C^ 
 
 
 •^a 
 
 as 
 
 
 * See Loney's Co-ordinate Geometery, page 16. 
 
 \ Students unfamiliar with trigonometry should omit exercise 11. 
 
12 DETERMINANTS. 
 
 IV. 
 
 DETERMINANTS OF ANY ORDER. 
 
 11. A determinant of « rows and n columns is called a 
 determinant of the nth order, n being any positive whole 
 number. 
 
 12. The convention for such an expression is that it rep- 
 resents the algebraic sum of all the products formed by 
 taking one and only one element from each row and one and 
 only one element from each column. 
 
 13. The elements of any constituent are so written that 
 the symbols representing the elements shall stand in the 
 order of the columns from which they are taken. The 
 numbers indicating the rows from which they are taken are 
 written in a horizontal line. 
 
 14. The Sign Symbol, of the constituent is (— 1) ", w being 
 the number of inversions of order in the line. 
 
 To explain what is meant by an inversion we will take the 
 determinant 
 
 '\ 
 
 c. 
 
 
 I2 0^ Cl^ O2 I 
 
 ^3 ^2 «! - C3 «2 '^1- 
 
 In the first three constituents of this development, the 
 subscripts are in their natural order; hence, there are no 
 inversions- 
 
 In the last three constituents the subscripts may be ar- 
 ranged in a line as follows: 
 
 3 2 1 
 
 Here 3 stands before 2 and 1, and 2 stands before 1; hence, 
 there are three inversions. 
 
 From this we derive the following 
 
 RULE FOR SIGNS. 
 
 If in the sign symbol (— 1)"' n be eve7i, the sign of the 
 constituent is plus; but if n be odd, the sign of the constit- 
 iient is minus. 
 
DETERMINANTS. 13 
 
 V. 
 
 THE NUMBER OF CONSTITUENTS. 
 
 15. The symbol for the number of constituents is |«.* 
 Hence, a determinant of the second order has 1.2, or 2 con- 
 stituents; one of the third orA&r has 1.2.3, or 6 constituents; 
 one of i\\e fourth order has .1.2.3.4, or 24 constituents; and 
 in general one of the nth order has 1.2.3. .{n~l)n con- 
 stituents. 
 
 It follows that any polynomial of \n' terms, or one that 
 can be reduced to \n^ terms, may be written in the form of 
 a determinant. 
 
 Thus, the binomial a ~ b is identical with the determinant 
 \a 11 ^ II 11 
 \b ll \b a\ and 
 a 4- b may be written i 1 ll 
 
 |— b a\' Also, 
 2 X — 2_)' may be written 1 2 y\ 
 
 |2 x[ 
 EXERCISE III. 
 Write the following polynomials in the determinant nota- 
 tion: 
 
 1. l + a"" + b"" + c^. 
 
 2. 1+a^ + b^ + c^ + d\ 
 
 3. x^y^ - x^y^ + x^y^ - x^y^ + x^y^ - x^y^. 
 
 4. abc + ^a-^b-^c^ — aa\ — bb\ — cc\. 
 
 5. (a-b) (b-c) (c-a). 
 
 6. a^d^ + b^e^ + c'^P - 2 beef- 2 cafd - 2 abde. 
 
 7. x^ ^y"-^ z''-2y''z^-2z^x^ -^'Ix^y^. 
 
 8. fl^-f- 5* -ht* -t-rf* -Ib^c^ -^c^a"" -2a''b^ -ia^'d^ - 
 2b^d^-2c^d^ -8abcd. 
 
 9. Show that any polynomial may be considered as com- 
 posed of I n terms. 
 
 *In modern books on algebra the symbol |« is sometimes written 
 n/ See Chrystals algebra. 
 
u 
 
 DETERMINANTS. 
 
 VI. 
 
 GENERAL PROPERTIES OF DETERMINANTS. 
 
 16. T/te value of a determinajit is not altered by changing 
 the successive rows into successive columns. 
 
 For ' ,M 
 
 \a„ b,\ 
 
 a^b^ — a^b^ 
 
 \bv b^ 
 
 Note. — This principle is evident from the law of forma- 
 tion, being perfectly symmetrical with regard to rows and 
 columns. 
 
 Verify the truth of this principle by developing the fol- 
 lowing determinants: 
 
 i y^ ^3 
 
 yx y-. 
 
 ys 
 
 17. Jf any two rows or colum.ns be interchanged, the sign of 
 the determi7iattt will be changed. 
 
 For this is equivalent to a single permutation among the 
 subscripts and this by the law of formation causes a change 
 of sign. 
 
 Verify this principle by developing the following determi- 
 nants: 
 
 b, 
 K 
 b. 
 
 b, 
 b„ 
 b\ 
 
 18. If two rows or columns are identical, the determinant- 
 vanishes. 
 
 For if we interchange these rows or columns, we ought 
 to have a change of sign by the preceding principle; but 
 the interchange of two identical rows or columns can produce 
 
DETERMINANTS. 
 
 15 
 
 no change in the value of the determinant; in other words, 
 the detenninant is equal to itself with its sign changed, but 
 this can only be when it is equal to zero. 
 
 Verify this principle by developing the following determi- 
 
 nants: 
 
 0, 
 
 
 c„ 
 
 19. If every element in any row or cilumn be multiplied by 
 the satne factor, the determinant is multiplied by that factor. 
 
 This appears from the fact that every constituent in the 
 development of the determinant contains one and only one 
 element from the same row or the same column. 
 
 Ve^'ify this by developing the determinant: 
 
 «1 
 
 K 
 
 n a^ 
 
 n b^ 
 
 «3 
 
 b. 
 
 n c. 
 
 bx 
 b. 
 b. 
 
 Note. — This principle is true whether n be positive or 
 negative, integral ox fractional. 
 
 20. If the elements in one row or column be like multiples 
 of those of another row or column respectively, the determinant 
 vanishes. 
 
 This follows directly from Articles 18 and 19. I^et the 
 student make the application. 
 
 Verify the principle by developing the following determi- 
 nants: 
 
 0, 
 
 m fflj 
 
 m 
 
 b. 
 
 m. c^ 
 
 
 
 a. 
 
 
 b. cA 
 
 «i 
 
 
 K 
 
 Cx 
 
 = m 
 
 «i 
 
 bi c^ 
 
 «2 
 
 
 b. 
 
 c^ 
 
 
 «2 
 
 bi c^ 
 
 
 2 
 
 3 
 
 4 
 
 
 2 
 
 3 
 
 4 
 
 
 
 4 
 
 6 
 
 8 
 
 = 2 
 
 2 
 
 3 
 
 4 
 
 = 0. 
 
 
 1 
 
 2 
 
 3 
 
 
 
 1 
 
 2 
 
 3 
 
 
16 
 
 DETERMINANTS. 
 
 VII. 
 MINORS AND CO=FACTORS. 
 
 21. If we delete any number of rows and the same num- 
 ber of columns of a determinant, the remaining elements 
 taken in order form a determinant called a minor. 
 
 22. The elements common to the deleted rows and col- 
 umns taken in order form a determinant which is also called 
 a minor. 
 
 23. These minors are complementary. 
 
 To illustrate these terms we will take the following de- 
 terminant and delete the first two rows and the first two 
 
 columns. 
 
 a, b^ c^ d^ 
 
 , b^ c„ I 
 
 , b^ c^ I 
 
 b. c. I 
 
 The complementary minors are 
 
 \ ^ ,' and ■' 
 
 d,\ 
 
 24. If only one row and one column are deleted, what 
 remains is called a principai, minor. 
 
 25. The element common to the deleted row and column 
 and the corresponding principal minor are called co-factors. 
 
 This leads to the following principle: 
 
 26. A determinant of the nth order may be expressed in 
 terms of determinants of the {n — V)th order. 
 
 Thus: 
 
 K 
 b. 
 
 
 K 
 
 ;:H-^. 
 
 \a„ 
 
 '^ \b\ 
 
 \K 
 
 1^ 
 
DETERMINANTS. 
 
 17 
 
 To accompHsli this reduction we have the following 
 RULE. 
 
 Multiply each element of the first row or column by its 
 co-factor, marking the products alternately plus and minus. 
 Then take the algebraic sum of the results. 
 
 27. If all the elements but one of any row or column of a 
 determinant •Oanish, the order of the determinant is reduced by 
 one. 
 
 For if in the determinant 
 
 a, 
 a„ 
 
 terminant becomes 
 
 b. 
 
 a„_ and a^ both vanish, the de- 
 
 1^3 C^ 
 
 28. Any determiyiant may be exhibited in the form of 07ie 
 of any higher order. 
 For the determinant 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 «1 
 
 b. 
 
 ^1 
 
 
 X 
 
 ^1 
 
 K 
 
 c 
 
 a^ 
 
 b-z 
 
 C-2 
 
 = 
 
 y 
 
 a.. 
 
 b^ 
 
 fj 
 
 a-z 
 
 K 
 
 Ci 
 
 
 z 
 
 «3 
 
 b. 
 
 r 
 
 by Art. 27, in 
 which X, y, and z have any values whatever; and this process 
 may be extended indefinitely. 
 
 Note. — The principles in the last two articles enable us to 
 reduce determinants of a higher to a lower order and the 
 converse. 
 
 29. If every element in any row or column be compound, the 
 determinant is resolvable into the sum of others. 
 
 For, if in the determinant 
 
 b. 
 
 we write a^+ k for a , , 
 
18 
 
 DETERMINANTS . 
 
 b^+l for 6^, c^+m ior c^, the determinant becomes 
 a„ b„ c^ 
 
 K+.) jj H-(^+/)£ ::K(^.^-)C, 
 
 The expansion of this establishes the principle. Let the 
 student perform the indicated operations. 
 Aliter. — Let us take the determinant 
 
 K 
 b. 
 
 b^ + c„ 
 + c. 
 
 dn 
 di 
 
 in which the elements 
 of the second column are compound. Developing this and 
 separating each constituent into parts, we have 
 
 + <2] (l^j + f„) (/g = + flj ^2 f/j + fli f„ 1/3 , 
 
 + a^ {b^ + (-3) fl?j = + i^a 1^3 <a?i + a^ c^ d^ 
 
 + a^ (b^ + fi ) fl?3 = + «3 ^j a?2 + «3 c, d^ 
 
 — ^3 (^3 +^2)^1 = — aj ^2 rf, — ftj c„ d^ 
 
 — a, (^j + f J ) a?3 = — a^ b-^ d^ — a^ c^ d^ 
 
 «i (^3 + -^3) ^2 = - «i bs "2 
 
 ^2 
 
 Summing these and using the determinant notation, we 
 have 
 
 d, 
 d. 
 
 a^ b^ + fj d-^ flj 5j d-^ a^ c^ 
 
 «3 b^ + c„ fl?2 = «2 b„ d„ + «2 '"2 
 
 «3 ^3 + ^3 </3 «3 ^3 <^3 (Zj ^3 
 
 which proves the proposition. 
 
 30. If the clcme7its of one row or coluinn are 7-cspcctively 
 equal to those of other rows or columns viulliplied respectively 
 by constant factors ^ the determinant vanishes. 
 
 For it is the sum of other determinants which are sepa- 
 rately evanescent. 
 
DETERMINANTS. 
 
 To illustrate take the determinant 
 
 k b^ + /f, by Cy 
 k b„ + lc„ b., c. 
 
 19 
 
 ^3 b. 
 
 k b-i + Ic. 
 which vanish by Article 20. 
 
 kb^ by Cj 
 
 
 Ic 
 
 kb^ ^j, (Tg 
 
 + 
 
 Ic 
 
 kb^ b.^ C3 
 
 
 Ic 
 
 K 
 
 31. A determinant is not altered if we add to each element 
 of any row or column the corresponding element of any other 
 row or column multiplied by constant factors. 
 
 To show this let us take the determinant 
 
 b,. 
 
 and add to the elements of the first col- 
 umn the corresponding elements of the second column mul- 
 tiplied by k and we obtain 
 
 a, + kby 
 
 b. 
 
 Cx 
 
 
 «1 
 
 b. 
 
 ^1 
 
 
 kby 
 
 bx 
 
 Cx 
 
 a^ + kb^, 
 
 b. 
 
 ^2 
 
 = 
 
 «2 
 
 b. 
 
 ^2 
 
 -V- 
 
 kb^ 
 
 b. 
 
 ^2 
 
 a 3 + kb^ 
 
 b. 
 
 ^3 
 
 
 «3 
 
 b. 
 
 Ci 
 
 
 kb. 
 
 b. 
 
 Ci 
 
 but the last determinant vanishes by Article 20, which estab- 
 lishes the principle. 
 
 EXERCISE IV. 
 Reduce the following determinants: 
 
 1 
 
 3 - 
 
 4 3 
 
 10 1 
 la 2 
 
 ,-1 
 
 1-1 21 13 21 _^ 13-11 _ 
 I 3 - ll ~ k - l| + k 3| ~ 
 (1_6) -(-3 -8) + (9 4-4) =19 
 
 AliTER. — [Using the principle in Art. 30.] Subtract the 
 elements of the first column from the corresponding elements 
 
20 
 
 DETERMINANTS. 
 
 of the second and third columns; this without altering the 
 value of the determinant reduces it to 
 
 3-4-1 
 4-1-5 
 
 equivalent to 1—4 — 11 
 
 1-1-51 
 
 which is 
 or 20 - 1 = 19. 
 
 3 4-5 
 4-5 3 
 
 5-3-4 
 
 31 - 4 14 
 
 5 14-51 
 5-3; 
 
 3(20 + 93-4 (- 16 - 15) - 5 (- 12 + 25) = 146. 
 
 3. 
 
 6 4- 
 
 - 5 
 
 ;i6 
 
 52 
 
 59 
 
 9-5 
 
 3l = 
 
 = 29 
 
 22 
 
 39 
 
 1-3- 
 
 -4' 
 
 1 1 
 
 
 
 
 
 32 4-5 
 
 18-5 3 
 
 2-3-4 
 
 - I22 39I = '-^ (^^^2^ - 1^^^) = ^460. 
 
 Note. — The ^ 3d is obtained from the 2d by adding to 
 the 2d and 3d columns respectively the elements of the 1st 
 multiplied by 3 and 4 respectively. 
 
 3 32-5 
 
 4 18 3 
 
 5 2-4 
 
 19 3 
 
 1-4 
 
 4. 
 
 - 16 
 
 4 31 
 
 5-41 
 
 14 91 
 
 5 1 
 
 2 [3(- 36 - 3) - 16 (-16 - 15) - 5 (4 - 45)] 
 2 [- 117 + 496 + 205] = 2 x 584 = 1168. 
 
DETERMINANTS. 
 
 21 
 
 5. 
 
 3 4 32 
 4-5 18 
 
 = 3 
 
 5 181 
 
 -3 
 
 181 
 21 
 
 -32 
 
 4-51 
 5-3 
 
 3 (- 10 + 54) - 4 (8 - 90) + 32 (- 12 + 25) = 876. 
 
 9 
 
 30 
 24 
 
 13 17 
 28 33 
 40 54 
 37 46 
 
 4 
 8 
 
 13 
 11 
 
 = 
 
 1 1 1 
 
 2 4 1 
 4 1 2 
 
 2 4 2 
 
 4 
 
 8 
 
 13 
 
 11 
 
 
 
 1 
 4 
 
 1 
 4 
 1 
 
 4 
 
 1 1 
 
 1 1 
 
 2 6 
 2 3 
 
 ' 
 
 1 
 
 2 2-1 
 4-3-2 
 2 2 
 
 
 - 1 
 
 o 
 
 1 
 
 
 2-1- 
 -3-2 
 2 
 
 1 
 2 
 1 
 
 = 
 
 4 
 
 — 7 - 
 
 
 -1 
 
 _2 
 
 
 2 = 
 1 
 
 
 
 4 
 
 -7 
 
 -1 
 
 -2 
 
 = -8- 
 
 7 
 
 = 
 
 - 
 
 15. 
 
 
 
 Note. — The second determinant is derived from the first 
 by subtracting from the elements of the Jirsi, second and 
 t/iird cohimns, twice, three times and four times the corre- 
 sponding elements of the last column. The remaining steps 
 are similar. 
 
 7. 
 
 a 
 
 b 
 
 c 
 
 
 
 
 ^ 
 
 
 
 c 
 
 a 
 
 b 
 
 = a 
 
 a 
 
 b - 
 
 - b 
 
 c 
 
 b 
 
 b 
 
 c 
 
 a 
 
 
 c 
 
 a 
 
 
 b 
 
 a 
 
 c \c 
 
 \b 
 
 a(a^ -bc)~b (ac- b"^) + c (c-= - ab) = a^ +b''+c^ - Zabc. 
 
 Moreover, since by Art. 31 the determinant is unchanged 
 in value written 
 
 - Zabc. 
 
 a + b 
 
 + c 
 
 b 
 
 c + a 
 
 + b 
 
 a 
 
 b +c 
 
 + a 
 
 c 
 
 : {a + b + c) 
 it is evident that {a + b + c') is a factor of a^ + b^ + C 
 
 1 
 
 b 
 
 c 
 
 1 
 
 a 
 
 b 
 
 1 
 
 c 
 
 a 
 
22 
 
 DETERMINANTS. 
 
 
 
 
 
 8. 
 
 
 
 
 
 
 
 a 
 
 b 
 
 c d 
 
 
 
 
 
 
 
 d 
 
 a 
 
 b c 
 
 '= a 
 
 a 
 
 b 
 
 c 
 
 -b d 
 
 b 
 
 c 
 
 c 
 
 d 
 
 a b 
 
 
 d 
 
 a 
 
 b 
 
 
 c 
 
 a 
 
 b 
 
 b 
 
 c 
 
 d a 
 
 
 c 
 
 d 
 
 a 
 
 
 b 
 
 d 
 
 a 
 
 d a c 
 
 -d 
 
 d a 
 
 b 
 
 c d b 
 
 
 c d 
 
 a 
 
 b c a 
 
 
 b c 
 
 d 
 
 a^ - b^ + c^ - d^ - 2 a^ c'i ^ 2 b^ d^ - A a'^ b d + i: b'^ a €- 
 ■i c'^ b d + i d'^ a c. And, as in example 7, it appears that 
 {a + b + c+ d) is a. factor of this determinant. 
 
 
 
 c b 
 
 c 
 
 a 
 
 b 
 
 a 
 
 9. 
 
 - c (0 - a 5) + 5 (a f - 0) = 2 a(5<:. 
 
 10. Show that the following determinant vanishes: 
 
 3 
 
 1 
 
 5 
 
 2 
 
 2 
 
 5 
 
 7 
 
 3 
 
 8 
 
 9 
 
 1 
 
 4 
 
 6 
 
 15 
 
 21 
 
 9 
 
 11. Prove the identity 
 
 be 
 
 a 
 
 a^ 
 
 
 ca 
 
 b 
 
 b^ 
 
 ^3 
 
 ab 
 
 c 
 
 c^ 
 
 
 12. Prove the identity 
 
 xys 
 
 V 
 
 -2 
 
 ^3 
 
 
 1 
 
 .■ 
 
 ysv 
 
 X 
 
 x" 
 
 X^ 
 
 
 
 1 
 
 X 
 
 Zv X 
 
 y 
 
 y' 
 
 y 
 
 
 1 
 
 y 
 
 vxy 
 
 3 
 
 z'^ 
 
 Z^ 
 
 
 1 
 
 z 
 
 yS 
 
 X* 
 
 y' 
 0* 
 
DETERMINANTS. 
 
 23 
 
 13. Prove 
 
 2 1-7 
 4-3 8 
 6 5-9 
 
 1 1 7 
 
 2 3 8 
 
 3 5 9 
 
 14. Reduce the following determinant to one in which all 
 the elements in the first row shall consist of units: 
 
 4 2 5 10 
 116 3 
 7 3 5 
 
 2 5 8 
 
 15. Prove the identity 
 
 {b — c) (c — a) i^a — b"). 
 
 16. Find the value of the determinant 
 8 7 2 20 
 
 4 
 
 11 
 
 
 
 i- 
 
 Ans. 2188. 
 
 17. Prove the following identity, and expand the determi- 
 
 nant: 
 
 1 
 
 1 
 
 1 2^ 
 
 1 y^ 
 
 1 
 
 
 x^ 
 
 
 Q X y 2 
 
 X 2 y 
 
 y 2 Q X 
 
 2 y X 
 
 18. Show that the following determinant vanishes: 
 
 d + c a 1 
 c + a b 
 a + b c 
 
 19. Find the value of the determinant 
 12 4 
 
 7 
 10 
 
24 
 
 DETERMINANTS. 
 
 20. Resolve into linear factors the determinant 
 
 abed 
 bade 
 c d a b ' 
 \d c b a 
 
 21. Find the value of the determinant represented by the 
 following- magic square: 
 
 4 9 2 
 
 3 5 7 
 
 1 
 
 22. Evaluate the determinant represented by the follow- 
 ing magic square: 
 
 1 Vh 14 4 
 
 12 6 7 9 
 8 10 11 5 
 
 13 3 2 16 
 
 23. Evaluate the following magic square: 
 
 10 18 1 14 22 
 4 12 25 8 16 
 
 23 6 19 2 15 
 17 5 13 21 9 
 
 11 24 7 20 3 
 
 24. The following determinant represents the ' ' Tour of 
 the Chess Knight," which is to pass over the entire chess- 
 board touching each spot but once, beginning on any square 
 and ending on any other square of an opposite color: Eval- 
 uate the determinant: 
 
 16 45 30 5 18 43 32 
 
 29 4 17 44 31 6 19 42 
 
 46 15 62 59 52 55 s 33 
 
 3 28 53 56 61 58 41 20 
 
 14 47 60 63 54 51 34 9 
 
 27 2 25 12 57 3H 21 40 
 
 48 13 04 37 50 23 10 3!". 
 
 1 26 49 24 11 36 39 22 
 
DETERMINANTS. 
 
 VIII. 
 
 EQUATIONS SOLVED BY DETERMINANTS. 
 
 32. The solution of simultaneous linear equations con- 
 taining two variables. 
 
 I,et us take the two general equations 
 
 flj X + b^y = /"j, (1) 
 a^x^ b^y = P^. (2) 
 
 Multiplying (1) by h., and (2) by i5, we have 
 
 a^ b^ X + b^ b„ y = P^ b^, (3) 
 a„ b^x + bi b„_y = P^ b^. (4) 
 
 Subtracting (4) from (3) we have 
 
 (rt, b^ - a^ b^) X = Pi b^ - P^ b^, 
 
 P. b^ ^ P„ b, 
 
 X = ~ ^^ — * 
 
 a, b^ — flj b^ 
 
 In a similar way or by symmetry we find 
 
 y = A ^i-A ^2 
 flj b^ — flj b^ 
 
 Writing these values of the variables in the determinant 
 notation we have 
 
 P. 
 
 p. 
 
 b. 
 
 a; 
 
 b, 
 
 ffg 
 
 *2 
 
 y 
 
 a, PJ 
 a,__Pj 
 a^ bA 
 a„ bA 
 
 This leads to the following rule for the solution of simulta- 
 neous linear equations containing two variables. 
 
26 DETERMINANTS. 
 
 RULE. 
 Either variable is equal to a fraction whose denominator 
 is the determinant formed by taking in order the co-efficients 
 of the variables, and whose numerator is the determinant 
 obtained from the denominator by replacing the co-efficients 
 of the required variable by the corresponding absolute terms 
 of the equations. 
 
 EXERCISE V. 
 
 Solve the following groups of equations by determinants: 
 
 1. 2. 
 
 nx + 7j = 
 
 67, 
 
 2x+7y- 
 
 = 41, 
 
 5x + 4y = 
 
 58. 
 
 8x + 4y: 
 
 = 42. 
 
 3. 
 
 
 4. 
 
 
 Ux + dy = 
 
 156, 
 
 ex+ iy 
 
 = 236, 
 
 7x + 2j = 
 
 58. 
 
 Sx + 15y 
 
 = 573. 
 
 5. 
 
 
 6. 
 
 
 39x + 27j)/: 
 
 = 105, 
 
 72x + 14y ■■ 
 
 = 830, 
 
 b2x+ 29j)/ : 
 
 = 133. 
 
 68x+ 7y: 
 
 = 273. 
 
 7. 
 
 
 8. 
 
 
 15x + 19y ■■ 
 
 = 132, 
 
 13x+17y. 
 
 = 189, 
 
 S5x + 17y: 
 
 = 226. 
 
 2x+ y . 
 
 = 21. 
 
 9. 
 
 
 10. 
 
 
 X + I'Sy = 
 
 = 49, 
 
 5 .r + Sy = 
 
 101, 
 
 3^-+ 7j' = 
 
 = 71. 
 
 9.r + 2r = 
 
 95. 
 
 11. 
 
 
 12. 
 
 
 2'dx-Uv -- 
 
 = 175, 
 
 171.r-213i 
 
 ' = 042, 
 
 S7x-my- 
 
 = 497. 
 
 114 r -320, 
 
 ■ = 244. 
 
DETERMINANTS. 
 
 13. 14. 
 
 12j;+ 7j/ = 176, 69j)/-17.r= 103, 
 
 3^-19x= 3. 14.;t;-13j' = -41. 
 
 15. 16. 
 
 2jc-7j'= 8, 
 
 4.r + 9j = 
 
 = 106, 
 
 4j/-9x = 19. 
 
 8x+ 17 J/ = 
 
 = 198. 
 
 17. 
 
 18. 
 
 
 17;c+12jj/= 59, 
 
 7«.r + nj 
 
 = e, 
 
 19^- 4j/= 153. 
 
 px +qy 
 
 = /• 
 
 19. 
 
 20. 
 
 
 6.r-5j/ = 39, 
 
 .ar + 2j/: 
 
 = 28, 
 
 7 .a: - 3 J/ = 54. 
 
 4.«--3^ = 
 
 = 24. 
 
 21. 
 
 22. 
 
 
 6.r-2j/= 2, 
 
 3;r + 2_y = 
 
 72, 
 
 2x + 3ji'= 19. 
 
 5;»r + 7_y = 
 
 175. 
 
 23. 
 
 24. 
 
 
 5x+7y = 201, 
 
 7.:c + 2j/ = 
 
 = 30. 
 
 8:«-3ji/= 137. 
 
 5 ;t + 3y = 
 
 = 34. 
 
 33. The solution of simultaneous linear equations con- 
 taining ikree variables. 
 
 Let us take the general equations 
 
 a^ X + 6^y + Ci^ s = P^, (1) 
 a^ x+ 6^j + c^ z = P^, (2) 
 «3 ^ + -^3 J'' + -^3 ^ = A ■ (3) 
 
28 
 
 DETERMINANTS. 
 
 Multiplying (1) by 
 
 b^ cA, (2) by - \b^ cA and 
 
 1^3 ^3 
 
 b^ cA 
 b„ c„\ 
 
 V = 
 
 (3) by 
 
 and adding the resulting equations mem 
 ber by member and reducing, we easily obtain 
 
 ^ ^ Pxbj^i- P-jb^c^ + P^b^c^ - P^b^c^ + P;b^c^ - P^b ^c^ 
 a^b^c^ — a^b^c^ + a^b^c^ — a^b^c^ + a^b^c^ — a-^b^c^' 
 
 In a similar way we find 
 
 a^P„Cy — a^P.^c^ + a^P^c^ — a^P^c^ + a^P^c^ — a^P^c^ 
 a^b^c^ — a^b^c^ + a^b^c^ — a^b^c^ + a^b^c^ — a^b^c^' 
 
 And also 
 
 ^ ai b^P.^~a^b .^ P^ + a^bsP^-a^b^P^ ^ a^b^P^- a^b^P^ 
 ''■ib^c^ — a^b^c^ + a^b^c^ — a^b^c^ -\-a^b^Ca — a^b^c^ 
 
 Writing these values in the determinant notation, we have 
 
 and z = 
 
 Px b, c, 
 P, b, c, 
 P. b, c. 
 
 . J^ = 
 
 «i Px ^1 
 
 "■% Pi ^2 
 «3 A ^3 
 
 «i b^ c^ 
 a^ b^ c^ 
 
 flg ^3 Cg 
 
 «3 1^3 -^3 
 
 k 
 
 b. 
 
 P, 
 
 'fls 
 
 bi 
 
 P. 
 
 '«3 
 
 b. 
 
 Ps 
 
 a, 
 
 b. 
 
 Cx 
 
 «2 
 
 b„ 
 
 (-„ 
 
 «3 
 
 bs 
 
 Ci 
 
 Note 1.— The student, in making the reductions for the 
 values of the variables, should use determinant methods. 
 Thus, in finding the value of x, the resulting coefficient is 
 
 bi cA - ^2 \b]_ c, I + ^3 l^i 
 
 1^3 C; 
 
 1*3 C^ 
 
 C„\ 
 
 The resulting coefficient of y is 
 
 bx 1^3 <^2| - b^ \b^ Cjl + ^3 i^j cA 
 
 1*3 <^-! 
 
 «1 *1 ^1 
 
 a 2 *o fo 
 
 a, bo <r. 
 
 *i *i c^ 
 
 *2 *2 -^2 
 
 *3 b, c. 
 
DETERMINANTS. 
 
 29 
 
 which, by Art. 18, is equal to zero, as it should be for the 
 purpose of elimination. In the same way it may be shown 
 that the resulting coefl&cient of z in the determinant notation 
 is equal to zero. 
 
 Notice also that the resulting value of the second member 
 of the reduced equation is 
 
 1 *. c. 
 
 - P^ b, c. 
 
 V P, b, c, = 
 
 P, b, c. 
 
 b-i c^ 
 
 bi c. 
 
 bi c^ 
 
 P, b, c, 
 A b, c. 
 
 Note 2. — After finding the value of one variable, we 
 may write the values of the other variables by the principle 
 of symmetry. 
 
 From the above solution it appears that the rule given for 
 the solution of equations of two variables applies also to the 
 solution of equations of three variables. 
 
 34. The solution of simultaneous linear equations contain- 
 ing fotir or more variables. 
 
 lyet us take the four general equations 
 
 ftj X ^ b^y + c^ z + d^ V = P^, 
 a^ X + b., y + i:„ z + d.^ v = P^, 
 «3 X + b^ y + C;^ z + d^ V = P, , 
 a^ X + b^j + c^ z + d^ V = P^. 
 
 (1) 
 (2) 
 (3) 
 
 and 
 
 multiply (1), (2), (3) and (4) in order by 
 
 b-i c„ d^ 
 
 
 b\ c^ d^ 
 
 
 b, c, d, 
 
 
 b, c^ d, 
 
 bi ^3 ^3 
 
 , — 
 
 bi ^3 ^3 
 
 , 
 
 b% C-l ^2 
 
 , and — 
 
 bi c„ d^ 
 
 bi ^4 ^4 
 
 
 bi c^ d^ 
 
 
 b^ c^ d^ 
 
 
 bi ^3 ^3 
 
 and find the values of the variables as before, and thus show 
 that the rule given in Article 32 holds true for the solution 
 of equations containing four variables. Let the student 
 finish the work, and also show the universality of the 
 method by finding the values of the variables in the foUow- 
 lowing five equations: 
 
30 DETERMINANTS. 
 
 a^ V + b^ w -\- c^ X + d^y + e^ z = P-i, (1) 
 
 a^ V + b^ w + c^ X + d^y -\- e,, z = P^, (2) 
 
 a^ V + b^ w + c.^ X + d^ y + e^ z = P3, (3) 
 
 a^v+b^w + c^x + diy + e^z = Pi, (4) 
 
 a^ v+ br^w + c.^ X + d^y ^- e^ z = Pr,. (5) 
 
 EXERCISE VI. 
 Solve the following equations by determinants: 
 1. 2.* 
 
 z „ ax + dy +gz = p, 
 
 X V 
 
 ^" + 3 + 4 ~ ""' bx + ey -\- hz = q, 
 
 X y_ z_ _ ex +/y+ iz = r. 
 
 3 + I "^ 5 " ' 
 
 X y z CO 
 4+1 + 6=38 
 
 3. 4.* 
 
 V ^ X + y + z = 10, av + £X + iy + mz - «., 
 
 V + X + y + 2z = li, bv + fx + jy + ti3 = li, 
 2v + 2x+y+ z = 13, cv + gx -^ ky + oz = y 
 
 V + X + 2y + ^^ + 13. dv + kx + Iy + pz = ' 
 
 5. 6. 
 
 x + 2y + iiz =17, X- y+ 2=30, 
 
 1/ +2^ +3;t:= 13, 8x-4_y + 22'= 50, 
 
 + 2.ar + 3j/ = 12. 27 x -Qy '+ ?, z = ^L 
 
 a. 
 
 z 
 
 X 
 
 6 
 
 5 +4 
 
 X 
 
 5 
 
 y z 
 + 4-3 
 
 X 
 
 4 
 
 3 +2 
 
 = 11, 
 
 = 35, 
 = 16. 
 
 X + fy + Iz = p, 
 mx + y + VIZ = q, 
 nx + ?ty + 2 = r. 
 
 * Solve (2) and (4) by observation. 
 
DETERMINANTS. 31 
 
 9. 10. 
 
 n+ V +x+y = 10, x-3y + 5z= 10, 
 
 u+v+x + 2= 11, 2y-^2 -vQu= 16, 
 
 u+v+y + 2=12, iis-5u + 7 V = 24, 
 
 u + x+y + z=13, 4m_6w+8x=-6, 
 
 v+x+y + z=14:. f) v-1 x+Qy = 36. 
 
 Note. — If any of the equations do not contain all the 
 variables of the group, write the missing variable in its 
 proper place with the coefHcient 0. 
 
 11. 12. 
 
 X y_ z_ ^ u + x+y = 6, 
 
 245 ^' u + X + z = 9, 
 
 y X -? _ n ti+y+z = 8, 
 
 3^ "•" 4' ■^ 10 " ^'" x+y + z = 1. 
 
 Z X V 
 
 — + — + — 
 3 5 6 
 
 1 Gi 
 
 18 0- 
 
 13. 
 
 2x + 2y+ ^=85-2m, 
 
 _y + iz= 31, 
 
 ix+ 2 = 32, 
 
 I + 3j/ = 23. 
 
 14. A boatman can row down stream a distance of 20 
 miles and back again in 10 hours, the current being uni- 
 form all the time; and he finds that he can row 2 miles 
 against the current in the same time that he rows 3 miles 
 with it. Required, the time in going and returning. 
 
 15. A and B can do a piece of work in 6 days, A and C 
 in 8 days, and B and C in 12 days. In how many days can 
 each do it alone ? 
 
32 
 
 DETERMINANTS. 
 
 IX. 
 
 ELIMINANTS. 
 
 35. When the number oi independent equations in a group 
 equals the number of variables the equations are said to be 
 Consistent. 
 
 36. When the number of equations in a group is one 
 more than the number of variables, the equations are not 
 consistent unless the relation between the coefficients of the 
 variables is such as to make one of the equations depend 
 upon the others, and the expression of this relation is called 
 
 the EtlMINANT. 
 
 The eliminant is sometimes called the Discriminant. 
 To find the eliminant let us consider the following equa- 
 tions: 
 
 a^ X + b^y + P^ =Q, (1) 
 a, x-l- ^a y + /*, = 0, (2) 
 0. (3) 
 
 O3 ;«■ + ^3 JK + P^ 
 
 From equations (1) and (2) we derive 
 
 -A 
 -A 
 
 b. 
 b. 
 
 b. 
 b^ 
 
 P. 
 
 «2 
 
 b. 
 b. 
 
 and 
 
 y = 
 
 P. 
 
 «i 
 
 P. 
 
 J\ 
 
 a^ 
 
 A 
 
 b. 
 
 «i 
 
 b. 
 
 bi 
 
 «2 
 
 b. 
 
 Substituting these values in (;i) and clearing of fractions, 
 we have 
 
 1^ PA 
 
 \b, pA 
 
 «1 Pi\+ P3 l«l '''li 
 
 
 
DETERMINANTS. 
 
 33 
 
 or 
 
 P. 
 P^ 
 P. 
 
 = (4) 
 
 which 
 
 is the required eliminant. 
 
 The first member of equation (4) is called the determinant 
 of the group of equations. 
 
 From this we derive the following 
 
 RULE. 
 
 The eliminant is formed by putting the determinant of the 
 group of equations equal to zero. 
 
 Note. — This rule applies to any n equations containing 
 (ra — 1) variables, n being any positive whole number. 
 
 EXERCISE VII. 
 
 Find the eliminants of the following groups of equations 
 and test the consistency of each group: 
 
 1. 
 
 2. 
 
 3. 
 
 x+y = Q, 
 
 2x- _y = 1, 
 
 2x-v= 1, 
 
 x+y=\S 
 
 + -.>' = 2, 
 
 5^-2_)/ = 4, 
 
 Zx-y = ^, 
 
 X —y = 6 
 
 f _9 
 
 3 _r + 7 = 9. 
 
 \x —y = 4. 
 
 ^= 9 
 
 y 
 
 5. Show that, if in the group 
 
 x+y = s, 
 X — y = d, 
 
 X 
 
 s + d 
 
 the equations are consistent. 
 
34 
 
 DETBEMINANTS. 
 
 X. 
 HOMOGENEOUS LINEAR EQUATIONS. 
 
 37. Homogeneous Equations have a variable in every 
 term multiplied by a constant factor; thus, a^x + b^y + 
 c^z = is homogeneous. 
 
 38. When the number of homogeneous equations equals 
 the number of variables, the values of the variables are 0, 
 and the equations vanish unless the relation between the 
 coefficients make one of the equations depend upon the 
 others, in which case we can find the values of the ratios of 
 the variables, from which we may find the values of all the 
 variables except one, to which we may assign any values 
 whatever. 
 
 I<et us take the following homogeneous equations: 
 
 a^x^-b.^y^c^s = Q\ 
 
 Ur^ X + 6^ J/ + c„ 2 = y . (1) 
 
 a^ X + 6^ y + c^ z = ) 
 
 If z is not equal to 0, we have by dividing by z 
 
 X 
 
 a. — h 
 z 
 
 y 
 
 + c, 
 
 V 
 b. - + r„ 
 
 ^b-y 
 
 c,=0 
 
 ^(2) 
 
 z z 
 
 These equations will be consistent when 
 
 3 ^3 
 
 = 0. (3) 
 
 Equation (3) expresses the condition that one equation of 
 group (1) shall depend upon the others, and is the elimi- 
 nant of group (2) and also of group (1). 
 
DETERMINANTS. 
 
 35 
 
 Note. — This principle is true of n homogeneous equa- 
 tions containing n variables. 
 
 39. When the number of homogeneous equations is one 
 less than the number of variables, we can find the values of 
 the ratios of the variables. 
 
 I^et us take the equations 
 
 we derive 
 
 flj X -\- b^ y ^- c^ ^ = 
 a^ X -'r b^ y -\- c„ ^^ = 
 
 •(1) 
 
 from which 
 
 y 
 
 «! - + i^l - = - fi, (2) 
 
 b. 
 
 y 
 
 c,. (3) 
 
 Solving (2) and (3), we have 
 
 X 
 
 c„ b.\ , y a, 
 
 — ? -\ and - = ~- 
 
 a, b,\ 2 \a. 
 
 a„ 
 
 b. 
 
 1>A 
 b. 
 
 Hence, 
 
 X 
 
 y 
 
 -"^1 
 
 K 
 
 ^ ^i 
 
 K 
 
 a, 
 
 -c, 
 
 a„ 
 
 -c^ 
 
 X : y : 3 : 
 
 ■ ■. h^i '^il 
 
 b. 
 
 ^1 
 bj 
 
 which may be written 
 
 X : y : z : : \b^ 
 
 \b„ c.,\ 
 
 c„l 
 
 l:[^'^ 
 
36 
 
 DETERMINANTS. 
 
 EXERCISE VIII. 
 Find the ratios of the variables in the following groups: 
 
 Qx-%y+12z = 
 2 .r + 4 J/ - 10 ^ = 
 
 
 X + y + 2 = 0, 
 mx + ny + oz = 0. 
 
 3. 
 
 
 4. 
 
 6x-4jj/-3^ = 0, 
 7x-<dy-^2 = 0. 
 
 5. 
 
 ;tr + 4 J' - 3 2 = 0, 
 5x— y — 2 = 0. 
 
 X + 
 x + l 
 10 X- 
 
 y + 
 iy- 
 
 y - 
 
 2 = 0, 
 
 62 = 0, 
 
 2=0. 
 
 6. Ascertain whether the following homogeneous equa- 
 tions are consistent: 
 
 3x + Ay - 52 = 0, 
 ix + 2y-Ss = 0, 
 5x -By + iz = 0. 
 
 7. Ascertain whether the following homogeneous equa- 
 tions are consistent: 
 
 7x-y+ 5z = 0, 
 
 12x+y-24:z = 0, 
 
 X - y -I- ll^' = 0. 
 
 8. Form three consistent homogeneous equations, using 
 as coefficients of the variables the elements of the following 
 determinant: 
 
 1 1-1 
 
 2 2-2. 
 14 3 
 
 9. Show that any determinant whose vabie is zero repre- 
 sents consistent homogeneous equations in number equal to 
 the order of the determinant. 
 
DETERMINANTS. 
 
 XI. 
 
 MULTIPLICATION OF DETERMINANTS. 
 
 40. Let Z?i and D^ denote two determinants, 
 product may be expressed by 
 
 Their 
 
 
 
 Now, let /?, 
 
 \a„ 
 
 71 
 
 (5, 1 and /?2 
 bJ 
 
 \x„ j'^l and 
 
 let P= D, X £>„ ^ a, b, - 1 
 
 -1 
 
 J'l 
 
 Taking the co-factors of this determinant, we have 
 
 «1 
 
 ^ 
 
 -1 
 
 a^ 
 
 ^ 
 
 
 
 
 
 
 
 .r 
 
 
 
 
 
 X 
 
 b^ - 
 
 - 1 
 
 -fls 
 
 ^ 
 
 X, 
 
 JKi 
 
 
 
 
 x^ 
 
 J^2 
 
 
 
 
 which may be written 
 
 1 
 
 x^ y^ 
 
 b. x. 
 
 --(a^b^-a^b^) \x^ yA 
 \x^ y„J 
 
 yi\ 
 
 from which we derive the following 
 
 RULE. 
 
 The product of two deiernii?ia7its viay be forvied by writing 
 a determinant of a higher order of which the two factors are 
 complementary m.inors, and filling vacant places due to one or 
 both factors with zeros, and filling the remaining places with 
 any finite elements. 
 
 41. The product of two determinants of any order is a 
 determinant of the same order. 
 
 I,et us find the product /^ of two determinants, /?, and D^. 
 
38 
 
 DETERMINANTS. 
 
 I,etZ?i 
 
 = 
 
 «1 ^1 
 
 Cl 
 
 and L 
 
 2 = 
 
 -^'i 
 
 X2 
 
 
 a^ 0^ 
 
 Ct 
 
 
 
 Jl J2 
 
 
 a 3 ^3 
 
 C3 
 
 
 
 2l ^2 
 
 Tlien by the rule we 
 
 have 
 
 
 P = 
 
 ttj 6i 
 
 c, 
 
 
 
 
 
 «s 62 
 
 t 
 
 r. 
 
 
 
 
 
 «3 ^3 
 
 I 
 
 -3 
 
 
 
 
 
 -1 
 
 C 
 
 ) X, 
 
 X2 x^ 
 
 
 
 -1 
 
 c 
 
 Ji 
 
 Vi yz 
 
 
 
 
 
 
 
 ~1 
 
 ^1 
 
 2i 
 
 ^3 
 
 
 Ji 
 
 In this determinant, add to the fourth column the sum of 
 the first multiplied by x^ , the second by jj/j , and the third 
 by z^ ; add to the fifth column the sum of the first multi- 
 plied by X2 , the second by j'2 , and the third by Z2 ; and add 
 to the sixth column the sum of the first multiplied by x^ , 
 the second by jj/3 , and the third by Zg . Then the determi- 
 nant becomes 
 
 a^ b^ c^ a^x^-\^b^y^+c^z^ a-^x^-^b^y^+c^z^ a^x^+b^y^+c^z^ 
 «, b^ c, a„Xj^-\-b^y^-\^c„z^ a„X2+b2y2+^2^2 <^ 2-'^' 3+^ 2^3+^2^; 
 3-^'i+'^3J'i+'^3^i a-jX^+b^^+^sZ^ a^x^+b^^+c^s.^ 
 
 •■3 ^i 
 
 -10 
 0-10 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 And this is equal to the product (with the proper sign) of 
 the determinant 
 
 1 
 
 
 
 
 
 
 
 -1 
 
 
 
 
 
 
 
 -1 
 
 (which is equal to — 1), 
 
 multiplied by the complementary minor or the determinant 
 of the third order 
 
 a,.T-j4- b^y^+c^3^ «i-f2+ b^y.,-\-c^z„ 
 a„x^+ b^y-^-V c^^i a^x^-l- b.;,y„+f„:::„ 
 a-^x^+b^i+c^z^ a^Xi+b0r„+c.^z„ 
 
 «l^'3 + '^lJ'3 + '^1^3 
 
 a„x,, + b.^^ + c.^z^ 
 
 '^■i-^'3 + l>3y3 + Cz33 
 
DETERMINANTS. 39 
 
 The formation of this determinant leads to the following 
 
 RULE.* 
 
 To form the determinant P, which, is the product of the two 
 determinants, D^ and D^ ; first, connect by plus signs the ele- 
 ments in the rows of both the determinants, Z>i and D^ ; sec- 
 ond, place the first row of D^ upon each row of D^ in turn 
 and let each two elements as they touch become products. This 
 is the first column of P. Perform the satne operation upon 
 Z?2 with the second row of Z?! to obtain the second column of 
 P, and agahi with the third row of Z?j to obtain the third 
 column of P, and so on. 
 
 EXERCISE IX. 
 1. Prove the identity 
 
 a. 
 
 ^ 
 
 <^1 
 
 Xi 
 
 yi 
 
 ^1 
 
 «2 
 
 b„ 
 
 ^2 
 
 x^ 
 
 J2 
 
 ^2 
 
 a,, 
 
 ^, 
 
 <^3 
 
 x^ 
 
 .r.s 
 
 ■^.■i 
 
 
 
 
 
 
 
 d. 
 
 <?i 
 
 /, 
 
 
 
 
 
 
 
 d. 
 
 e?. 
 
 /. 
 
 
 
 
 
 
 
 d. 
 
 ^3 
 
 A 
 
 
 «i b^ c. 
 
 
 dx ^1 /l 
 
 = 
 
 a^ b^ c.^ 
 
 
 d, e, /, 
 
 
 a% bi ^3 
 
 
 ^3 ""3 /s 
 
 2. Prove the following expression for the square of the 
 determinant of the third order: 
 
 b. 
 
 h ^3 
 
 a^c„-^ a^c.^— 2^j/'., 
 
 ajfj+flj^i-S/'i/^j a^c^-^a^c^-'-^b.^b^ 
 
 a^c.;,+ a2C^ — 2b^b2 a^c^-ha^Cj~2b^b^ 
 2{c!„c„—bl) a„c^-\-a^C2 
 
 2(^3^3-^1) 
 
 h.,b. 
 
 *See " Carr's Synopsis of Pure Mathematics," Article 570. 
 
40 DETERMINANTS. 
 
 Hint. — Multiply together the two determinants 
 
 only by the factor 2. 
 
 K 
 
 C-i. 
 
 
 ^1 
 
 -26, 
 
 «i 
 
 
 K 
 
 ^2 
 
 
 ^2 
 
 -2^3 
 
 «2 
 
 
 h 
 
 Ci 
 
 
 ^3 
 
 -26, 
 
 «S 
 
 which diflFer 
 
 3. Find the product of 
 
 \x + w y + s\ 
 \a + d 6 + c\' 
 
 \x — w y ■ 
 \a — d 6 - 
 
 ■ z\ 
 
 4. Multiply 
 
 X y ^'l by 
 z X y\ 
 y z x\ 
 
 c 6 
 a c 
 6 a 
 
 5. Multiply 
 
 / 
 
 a 
 
 b 
 
 by L 
 
 a 
 
 m 
 
 c 
 
 c 
 
 b 
 
 c 
 
 n 
 
 
 6. Multiply 
 
 12 2 
 2 5 3 
 116 
 
 by 
 
 3 11 
 
 12 4 
 2 3 5 
 
 7. Square the determinant 
 
 8. Express 
 
 1 2 1 
 3 12 
 
 2 3 4 
 
 in the form of a determinant. 
 
 9. Multiply together the following determinants: 
 
 1 4 
 
 
 2 4 6 
 
 2 6 5 
 
 
 5 2 1 
 
 3 2 1 
 
 
 13 2 
 
 2 4 2 
 
 
 1 8 2 
 
 H 3 
 
 
 2 6 8 
 
 6 5 
 
 
 5 9 4 
 
DETERMINANTS. 
 
 41 
 
 XII. 
 
 RECIPROCAL DETERMINANTS. 
 
 42. A Reciprocal Determinant has for its elements 
 the complementary minors of the corresponding elements of 
 the original determinant, and is equal to its (n — l)th power. 
 
 Let the reciprocal of D be denoted by Z?, , and multiply 
 the two determinants 
 
 la, b. 
 
 . D, 
 
 A, 
 
 B, 
 
 c. 
 
 A, 
 
 B, 
 
 Q 
 
 A, 
 
 B, 
 
 Q 
 
 All the elements of the resulting determinant except those 
 in the principal diagonal vanish; hence, we have 
 
 DD^ 
 
 Therefore, D, = D"" 
 
 D 
 
 
 
 
 
 
 
 D 
 
 
 
 
 
 
 
 D 
 
 = D^ 
 
 EXERCISE X 
 1. Show that the determinant 
 
 1 1 3 
 3-1 1 
 1 -3 -1 
 
 the reciprocal of 
 
 1 1 2 
 
 2 1 1 
 1 2 1 
 
 and that it is equal to 16. 
 2. Find the reciprocal of the determinant 
 
 6 
 
 2 
 
 1 
 
 1 
 
 3 
 
 4 
 
 5 
 
 7 
 
 2 
 
42 DETERMINANTS. 
 
 3. Find the reciprocal of the determinant 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 1 
 
 3 
 
 6 
 
 10 
 
 1 
 
 4 
 
 10 
 
 20 
 
 4. Write the reciprocal of the determinant 
 
 5. Square the determinant 
 
 a 
 
 h 
 
 g 
 
 h 
 
 b 
 
 f 
 
 g 
 
 f 
 
 c 
 
 2 2 
 1 2 
 
 3 1 
 
 result is its reciprocal. 
 
 and thus show that the 
 
 6. Form the reciprocal of 
 
 
 a e 
 
 e b 
 
 f g 
 h i 
 
 f 
 
 g 
 c 
 
 / 
 
 h 
 i 
 
 J 
 d 
 
 ciprocal of 
 
 
 
 A a 
 -a B 
 
 - b -e 
 -c -f - 
 
 b 
 e 
 C 
 
 - h 
 
 c 
 
 f 
 h 
 D 
 
 -d 
 
 - g ~ 
 
 i 
 
 -J 
 
 g 
 
 i 
 
 J 
 E 
 
 8. Show that the product of two reciprocal determinants 
 is the reciprocal determinant of the product of the two 
 original determinants. 
 
DETERMINANTS. 
 
 43 
 
 XIII. 
 
 SYMMETRICAL DETERMINANTS. 
 
 43. Conjugate Elements of a determinant occupy simi- 
 lar positions on opposite sides of the principal diagonal. 
 Therefore, any two conjugate elements are situated in a line 
 perpendicular to the principal diagonal, and at equal dis- 
 tances from it on opposite sides. 
 
 44. A Symmetrical Determinant is one in which every 
 two conjugate elements are equal to each other. 
 
 The determinant 
 
 a h 
 
 g 
 
 h b 
 
 f 
 
 g f 
 
 c 
 
 is symmetrical. 
 
 45. The principal diagonal is called the Axis of Symme- 
 try, and the determinant is said to be axi-symmetric. 
 
 46. If all the elements of the axis of symmetry are zeros, 
 the determinant is said to be zero-axial. 
 
 EXERCISE XI. 
 
 1. Show that the reciprocal of a symmetrical determinant 
 is symmetrical. 
 
 2. Form the reciprocal of the symmetrical determinant 
 
 / 
 
 a 
 
 b 
 
 
 a 
 
 m 
 
 t 
 
 
 b 
 
 c 
 
 n 
 
 
 3. Form the reciprocal of 
 
 a 
 
 I 
 
 m 
 
 n 
 
 I 
 b 
 k 
 g 
 determinant is a sym- 
 
 k 
 c 
 h 
 
 4. Prove that the square of 
 metrical determinant. 
 
 5. Show that the product of two reciprocal determinants 
 is the reciprocal determinant of the product of the two origi- 
 nal determinants. 
 
44 
 
 DETERMINANTS. 
 
 XIV. 
 
 SKEW DETERMINANTS. 
 
 47. A Skew-Symmetric Determinant is one in which 
 every element is equal to its conjugate with the sign changed. 
 
 The determinant 
 
 
 
 a 
 
 b c 
 
 
 - a 
 
 
 
 d e 
 
 
 - b 
 
 -d 
 
 / 
 
 
 — c 
 
 — e 
 
 -f 
 
 is a skew-symmetric. 
 
 48. Since the elements of the principal diagonal are their 
 own conjugates, it follows that skew-symmetric determi- 
 nants are zero-axial. 
 
 49. A Skew Determinant is one in which all the ele- 
 ments except those of the principal diagonal are equal tp 
 their conjugates with the sign changed. 
 
 The determinant 
 
 a 
 - / 
 
 I 
 b 
 
 n p 
 k q 
 
 — in 
 
 -k 
 
 c r 
 
 - P 
 
 -q 
 
 -r d 
 
 is a skew determinant. 
 
 Note. — While a skew-sjmmetric deteTmma.nt is zero-axial . 
 a skew determinant is not. 
 
 50. A skew-symmetric determi7iant of odd order vanishes. 
 Verify this by evaluating the skew determinant 
 
 
 
 - / 
 
 I 
 
 
 
 m 
 n 
 
 — m 
 
 - n 
 
 
 
 51. A skew-symmetric determinant of even order is a per- 
 fect square. 
 
 Verify this by evaluating the skew determinant 
 
 
 
 I 
 
 m 
 
 n 
 
 m - 
 
 
 ^ P 
 
 P 
 
 
 9 
 r 
 
 n - 
 
 - q 
 
 ~r 
 
 
 
UKTKRMINANTS. 
 
 45 
 
 
 
 a 
 
 b 
 
 r 
 
 — a 
 
 
 
 d 
 
 c 
 
 - b 
 
 </ 
 
 
 
 f 
 
 - '- 
 
 c 
 
 -/ 
 
 
 
 EXERCISE XII. 
 1. Show that the skew-syiiuuetvic detenuinaut 
 
 = (a/"- /^-+ cJy~. 
 
 2. Express in powers of .r the skew determinant 
 
 .1" III n />' 
 
 — Ill X q r 
 
 — n — q X ,v 
 
 — p — r — s \ 
 
 'A. Evahwte the skew-sj-mmetric determinant 
 
 a- ab a A 
 
 o b b- br 
 
 — a f — be ('-' 
 
 MISCELLANEOUS EXERCISE XIII. 
 
 1. Write the equation 
 
 abc + -Xi;/i - ti/- - bt;- - c/i- = 
 in the fonn of a detenninant. 
 
 2. Find the values of x in the equation 
 
 .1 .1 
 
 X 
 
 •2 X 
 
 o o 
 
 ,r 
 
 0. 
 
 i>. Solve b\- determinants 
 
 ■2x+ i+3r=i;, 
 3.+ ^=5. 
 
46 
 
 DETERMINANTS. 
 
 4. Eliminate x, y, z from the equations 
 
 ax ^- hy + fz — / = 0, 
 hx + by -\- es — m = 0, 
 fx + ey + cs — n = 0, 
 Ix + my + ns "^ 
 
 0. 
 
 5. Eliminate x, y, z from the equations 
 
 a^ 
 
 b^ 
 
 c^ 
 
 
 
 + -^ + 
 
 z^ ~ 
 
 0, 
 
 x^ 
 
 y^ 
 
 
 al 
 
 bni 
 
 en 
 
 
 ^ 
 
 + ^; + 
 
 T^ 
 
 0, 
 
 y- 
 
 
 al'' 
 
 bm^ 
 
 cn^ 
 
 
 x^ 
 
 + — 2" + 
 
 z^ 
 
 0. 
 
 6. If 
 
 L - ax + by + dz 
 M = bx + cy + ez 
 N = dx + ey + fz 
 P = Ix -\- my + 7iz and 
 
 ■H = 
 
 a b d 
 bee 
 d e f 
 
 Prove that 
 
 I m n 
 
 L, a b d 
 
 M b c e 
 
 N d e f 
 
 P H\ 
 
 7. Show that (x+y+z) (x-y-z) (y-z-x~) (z — x -y) 
 
 8. Express 
 
 J2 
 J'3 
 
 X y z 
 
 X z y 
 
 y z X 
 
 z y X 
 
 in the form of a determinant. 
 
DETERMINANTS . 
 
 9. Show that the determinant 
 
 a 
 
 b 
 
 c 
 
 d 
 
 b 
 
 a 
 
 d 
 
 c 
 
 c 
 
 d 
 
 a 
 
 b 
 
 d 
 
 c 
 
 b 
 
 a 
 
 is divisible by {a^by - (c+dy, and by (a-by-{c-dy. 
 
 10. Write 2xy2 in the form of a determinant. 
 
 11. Write 2 abc(a + b + c)^ in the determinant form. 
 
 -15. 
 
 13. Prove that the condition, that ax^ + 2 bxy + cy^ should 
 contain Lx-\-My as a factor, can be expressed thus: 
 
 12. Prove that 
 
 9 
 
 13 
 
 17 
 
 4 
 
 
 18 
 
 28 
 
 33 
 
 8 
 
 
 30 
 
 40 
 
 54 
 
 13 
 
 
 24 
 
 37 
 
 46 
 
 11 
 
 c 
 M 
 
 L 
 
 M 
 
 
 = 0. 
 
 14. Prove that 
 b + c 
 
 a 
 
 b 
 c + a 
 
 c 
 a + b 
 
 a 
 
 b + c 
 c + a 
 
 c 
 
 a+b 
 
 a 
 b + c 
 
 b 
 c + a 
 a + b 
 
 2(a + b + cy 
 {b + c]{c+a){a + b)' 
 
 15. If «, ;3, V, « are the roots of the equation, 
 X*' — px^ + qx^ — rx + s = Q, 
 express, in terms of the coefficients, the determinant 
 
 a 1 1 1 
 
 1^11 
 
 1 1 7 r 
 
 Ills 
 
48 
 
 DETERMINANTS. 
 
 16. Prove that the determinant 
 
 a 
 
 b 
 
 c 
 
 c 
 
 a 
 
 b 
 
 b 
 
 c 
 
 a 
 
 is divisible by a + ^^ + ^^f, when "> is a cube root of unity. 
 
 17. Raise a + m — n to the second power and write the 
 result as a determinant. 
 
 18. Find the third power oi a + b - c and write it as a 
 determinant. 
 
 19. Arrange the nine Arabic digits as a determinant in as 
 many ways as possible, and evaluate each determinant 
 formed. 
 
 The following are examples of determinants in which the 
 elements are themselves determinants: 
 
 20. 
 
 m n 
 
 n p 
 
 
 n p 
 
 q ) 
 
 = / 
 
 11 p 
 
 P j 
 
 
 q j 
 
 j k 
 
 
 21. 
 
 p 
 J 
 
 i 
 
 k 
 
 n 
 
 q 
 
 n 
 
 q 
 
 m 
 
 n p 
 
 
 q 
 
 m 
 
 
 q i 
 
 
 j 
 
 n 
 
 
 
 
 = j 
 
 p j 
 
 j 
 
 k 
 
 
 j k 
 
 
 n 
 
 q 
 
 
 P 
 
 j 
 
 n 
 
 J 
 
 k 
 
 q 
 
 n 
 
 q 
 
 m 
 
 k 
 
 q 
 
 q 
 
 m 
 
 j 
 
 k 
 
 
 q 
 
 m 
 
 j 
 
 n 
 
 n 
 
 q 
 
 
 q 
 
 ni 
 
 m 
 
 n 
 
 n 
 
 P 
 
 j 
 
 n 
 
 11 
 
 P 
 
 q 
 
 ] 
 
 j 
 
 k 
 
 n 
 
 P 
 
 p 
 
 j\ 
 
 n 
 
 q 
 
 q 
 
 J 
 
 J 
 
 k 
 
 1 
 
 the square of 
 
 p 
 
 j 
 
 n 
 
 J 
 
 k 
 
 q 
 
 11 
 
 q 
 
 in 
 
j