444 
A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. 
[357 
viz., this is 
a, 
h, 
9> 
l 
+ ad — l 2 + be —f 2 + 2 {nh — mg) +1 
h, 
b, 
A 
m 
9’ 
f> 
C, 
n 
l, 
m, 
n, 
d 
13. The expression for (T — i2) -1 is obtained from that of (T + i2) -1 by merely 
transposing the terms of the matrix, or, what is the same thing, by changing the 
signs of fi, v, p, a, r. And it would be easy by means of these developed values 
to verify the foregoing comparison of (T — il) (T +12 ) -1 and (T +12) (T — i2) -1 . 
Article Nos. 14 to 22. Second Investigation. 
14. I consider from a different point of view the theory of a matrix 
a, 
b, 
C, 
d ) such that II -1 = ( 
P> 
l, 
-h, 
-d), 
e > 
A 
9> 
h 
o , 
k, 
~9> 
— c 
i , 
A 
k, 
l 
- n , 
-A 
A 
b 
m, 
n, 
o, 
P 
— m, 
- i, 
e, 
a 
or, as we may call it, a Hermitian matrix. 
15. Lemma. The determinant 
a, 
b, 
C, 
d 
e , 
A 
9> 
h 
i , 
A 
k, 
l 
m, 
n, 
o, 
p 
may be expressed, and that in two different ways, as a Pfaffian. 
16. In fact multiplying the determinant into itself thus, 
V 2 = 
a, 
b, 
C, 
d 
tr. 
d, 
c, 
-b, 
— a 
e , 
A 
9> 
h 
h, 
9> 
— e 
i , 
A 
k, 
l 
l, 
k, 
-A 
— i 
m, 
n, 
o, 
P 
P> 
o, 
-n, 
— m 
we find 
(d, c, — b,— a), (h, g, -/, - e), (l, k, -j, - i), (p,o,~ n, - m) 
2 = (a , b, c, d) 
» 
» 
» 
yy 
= 
¿>11 > S u , S 13 , 5 14 
(e , f, g, h) 
» 
» 
y> 
yy 
S 21> ¿>22, ^23) ^24 
(* > A k > 0 
)■) 
>> 
yy 
» 
^31) S 32> ®33> S34 
(m, n, o, p) 
„ 
„ 
yy 
S 41j ®42) ^43) ¿>44