32 



PEOFESSOE CATLET ON THE THEOET OP MATEICES. 



viz. #,i = (a, e, i, m^^m, i, — e, — a), &c. ; this is likewise a skew determinant, and we 

 have 



or extracting the square root of each side, and determining the sign by the comparison 

 of any single term, we have 



V :;=rj2 ^34"T*13 'aiT'^u ^23» 



which is the other of the required forms of V. 

 1 8. Consider now the maticix 



(a, h, c, d ) 

 e , f, g, h 



J 



k, I 



which is such that 



m, n, 0, p 



(a, h, c, d )-' ={ p, 

 e , f, g, h 



* , J. 



1c, I 



this gives; 



( ], 0, 0, )= 



0, 1, 0, 



0, 0, 1, 



0, 0, 0, 1 



m, n, 0, jp 



I, —h, ~-d ) 

 , Ic, ^g, —c 



— w, — i, e, a 



which is in fact 



a, i, c, d )( p, I, —h, —d) 



e , fy g, h , k, —g, -c 



i , j, Jc, I —n, —J, f, h 



m, n, 0^ p —w, —i, e, a 



(p, 0, — n, — m), (I, k, —J, —i), (—h, —g,f, e), {—d, —c, 5, a 



3s=(«, b, e, d) 



(e, f, g, h) 



{i , j, h I) 



(m, n, 0, p) 



( 1, 0, 0, )=( s,„ 5,3, 



0, 1, 0, 

 0, 0, 1, 

 0, 0, 0, 1 



^«5 ^43 5 



"125 



^22 5 



^32 5 



■*~*42 5 



— S.l )5 



— S2I 

 ~"*3I 



— 5., 



and the two matrices will be equal, term by term, if only 



i=Si4=;:S23, 



that is, if six conditions are satisfied^ 



