474 



REPORT 1878. 



1) Developing the determinant of the left member, the following identity 

 becomes immediately proved : — 





y» y* 



lb & 



Thus the theorem holds for determinants of the second order. 

 2) Suppose the theorem hold for determinants of the (n — l) st order, i.e., 

 pose that 



sup- 



(6) 



then it shall he proved that 



A^D.a; 



A = DA. 



Now, Aj being a certain minor of A, we denote by A v A 2 , ..., A„ the minors of A 

 which correspond to the constituents of its first column. Then, by a known pro- 

 perty of determinants, we have the identities: 



QjAj + /3jA x + yi A 3 + ..., + , Kl A„ = A, 

 a.jAj + 3 A. 2 + 7.1A3 + ,..., + KoA,i = o, 

 a 3 \ + /3 3 A 2 + y 3 A 3 + ,..., + k 3 A„ = 0, 



a„A, + /3„A 2 + y, ( A 3 , + ,...+ *„ A„ = 0. 



Multiplying these equations by «,,«._>, a 3 ,...,a„ respectively, and adding them, we 

 obtain the first of the next following equations. In the same manner, the others 

 are obtained by using as successive multipliers the constituents of every one of the 

 other rows in D. By the notations (1) we thus get : 



(«n)Aj, + («0)A O + («y)A 3 + ,...,+ (W)A» = «jA, 

 (6a) Ai + (6/3) A 2 + (6y)A 3 + ,..., + (b K )A n = 6, A, 

 (ca)A, + (c/3)A 2 + (cy)A 3 + ,...,+ (a<)A n = C 1 A, 



(ka)A 1 + (/c/3)A 2 + (/^y)A 3 + ,..., + (7«)A„ = k,A. 

 Now eliminating A.,, A 3 , ..., A n between these equations, and putting 



«„ (aB), (ay), ..., (a K ) 

 K W), (by), ..., (b K ) 

 c» (c8), (cy), ..., (ck) 



k t , (k8),(ky), ..., (Jck) 

 we get 



(6) . . . _ . . . A . A : = v • A, 



Subtracting, in the expression for v, the constituents of its first column multiplied 

 by /3j from the corresponding constituents of the second column, multiplied by y x 

 from those of the third column, and so on, and finally multiplied by k 1 from 

 those of the last column, we get by a known property of determinants and in 

 virtue of (2) 



«i, («0)i, («y)i, •-, (««)i 

 c v ( c /3)i, (cy)„ ..., (ck), 



(7) 



K (kp)iAPy)i, —, (*<0i 



Now, D, being a certain minor of D, we denote by D 1} D 2 , ..., D„ the minors of 

 D, which belong to the constituents of its first column. Then we have the well- 

 known equations : 



aj), + 6jD 2 + cj) 3 + ... + £,D„ = D, 



(8) 

 and 



(9) 



"cUD; + 6 g D 9 + c 2 D 3 + ... + kj) n = o, 

 a 3 D 1 + 6 3 D 2 + e 3 D 3 + ... + & 3 D„ = o, 



.«„D 1 *6 b D 8 + Cb D 3 + 



k,P„ = o. 

 The first of the next following equations is obtained by addition of the equations 



