446 
A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. 
[357 
which is such that 
( 
a 
b, 
c, d ) _1 = ( 
V y 
l, 
- h, 
e 
f, 
9, h 
O , 
k, 
-9> 
i 
j > 
k, l 
-n, - 
1 y 
fy 
this 
gives 
TO 
n, 
o, p 
— TO, — 
i, 
e, 
( 
o, 
0, 
0 
)=( 
a , 
b, 
c, d )( 
V 
l, - 
-h, 
-d 
0, 
1, 
o, 
0 
£ 
f, 
g, h 
0 
k, - 
-g> 
— c 
0, 
0, 
1, 
0 
% 
> 
3 > 
k, l 
— n 
~ j > 
fy 
b 
0, 
o, 
0, 
1 
TO, 
n, 
o, p 
— TO 
-i, 
e, 
a 
(p, 0 
-n, - 
to), (l, k, - 
~jy 
-*)» ( 
= (a , b, c, d) 
(e , f, 9, h) 
(i , j, k, l) 
(to, n, o, p) 
which is in fact 
( 1, 
o, 
0, 
0 
)=( 
^14 > 
$13 J 
$12 > 
-«11 ) 
0, 
1, 
o, 
0 
$24 > 
$23 y 
$22» 
$21 
°, 
o, 
1, 
0 
$34 J 
$33> 
$32» 
~ $31 
o, 
0, 
o, 
1 
$44 , 
$43 > 
$42) 
$41 
and the two matrices will be equal, term by term, if only 
1 = = S 23 , 0 = 5 13 = S 12 = s. u = S 34) 
that is, if six conditions are satisfied. 
19. But we have also (a matrix and its reciprocal being convertible) 
( 1» 
0, 
o, 
0 
)=( 
P y 
ly 
-h, 
-d 
)( 
a , 
b, 
c, d 
0, 
1, 
0, 
0 
0, 
k, 
-g> 
— c 
e , 
fy 
g, h 
0, 
0, 
1, 
0 
— 11, 
~jy 
fy 
b 
i , 
jy 
k, l 
0, 
0, 
0, 
1 
— TO, 
- i, 
e, 
a 
TO, 
11, 
o, p 
(a, e, i, to), (b, /, j, n), (c, g, k, o), (d, h, l, p) 
( 
p, 
l, 
- h, 
-d) 
( 
0 , 
k, 
~9y 
-c) 
( - 
- 11, 
~jy 
fy 
b) 
( - 
- TO, 
- i, 
e, 
a) 
))