440 A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. [357 
5. I write for greater convenience 
- n = H- 1 (T - il) (T + il)- 1 ii, 
- n _1 = il” 1 (T + il) (T — il)" 1 ii, 
and I compare in the first instance the matrices (T — il) (T + il) -1 and (T + il) (T — il) -1 . 
6. Any matrix whatever, and therefore the matrix (T + il) -1 , may be exhibited as 
the sum of a symmetrical matrix and a skew symmetrical matrix; that is, we may 
write 
(T + n)~' = T + il', 
where T' is symmetrical, il' is skew symmetrical. We have then 
(T + il) (T + il)- 1 = (T + il) (T' + ii'), = 1, 
where, here and in what follows, 1 denotes the matrix unity. Moreover 
T — il = tr. (T + il), 
and thence 
(T - O)- 1 = (tr. (T + il))“ 1 = tr. (T + H)- 1 = tr. (T + W) = T - O'; 
that is 
(T - il)" 1 = T' - il'; 
and thence also 
(T - il) (T - il)" 1 = (T - il) (T - il') = 1. 
We have therefore 
(T - il) (T + il)- 1 = (T + il — 2il) (T' + il') = 1 - 2il (T + il'), 
(T + il) (T - il)“ 1 = (T - il + 20) (T - il') = 1 + 2il (T' - il'). 
7. Suppose for a moment that 
T' + il' = ( a , b, c, d ) 
e, f, g, h 
i , j, k, l 
m, n, o, p 
T' - il' = ( a, e, i, m ). 
/, j> n 
c, g, k, o 
d, h, l, p 
and therefore