PEOPESSOE CATLEY ON THE THEOEY OF MATEICES. 



33 



19. But we have also (a matrix and its reciprocal being convertible) 

 ( 1, 0, 0, )=( p, I, -h, -d {a, b, c, d) 



0, 1, 0, 

 0, 0, 1, 

 0, 0, 0, 1 



, h —g, —c 

 — m, — i, e, a 



e , f, g, h 

 i , j, k, I 

 m, 71, 0, p 



--( jp, I, —h, —d) 



( , k, —g, —c) 



(-n, -J, f, b) 



{—m, —i, e, a) 



which is in fact 



( 1, 0, 0, )=( #,, t,,, t,,, f,, 



^13 > ^23 > ^33 5 ''43 



^^125 "225 ^32? ^42 



''m ^21 » '^31? ^^41 



0, 1, 0, 

 0, 0, 1, 

 0, 0, 0, 1 



and we obtain for the equality of the two matrices the six conditions 



J-— -^14-— '235 "— -^13— -^12^'24-— f34 5 



equivalent to the former set of six conditions. 



20. We obtain from either set of conditions, for the determinant the value 



a , b, c, d 



^ t f, g, h 



^ ■> J i k, I 



m, n, 0, p 



= K 



21. Write 



(x,y,z,w)=i a, b, c, d XX,Y,Z,W); (w',y',z',w')=i a, b, c, d XX', Y', Z', W), 



e ,f, g, h 

 i , j , k, I 

 m, n, 0, p 



e , f, g, h 

 i , j , k, I 

 m, n, 0, p 



then substituting for (x, y, z, io){a/, y', 2', w') their values, we find 



Tw'+yz'-zy'-wc(f=-{t,„ t,„ t,„ i^., JX, Y, Z, W^X', ¥', Z', W) 



^315 ^22 5 ^23 5 ^24 



^315 '32 5 ^33 5 ^34 



'41 > ^42 > ^43 5 '44 



MDCCCLXVI. p 



