357]
A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES.
443
12. I remark in reference to the foregoing proof that writing
T= ( a, h, g, l )
h, b , f, m
9, f, c, n
l, m, n, d
then the actual value of
(T + il)-\ = ( a , h , g , l -1 )- x
h , b , f— 1, m
9 , f+1, c , n
l +1, m , n , d
_ 1 ( A. + d , H + w + v, G — Td — ya, L — l + p )
H + n — v, B + c , F —f + M — g + a
G-m + g, F -/- C+b , N + h + r
L — l — p , M — g — a, N + h — T , D +a
where
is
(A,
H,
G,
L
)
H,
B,
F,
M
a,
F,
c,
N
L,
M,
N,
D
t minors
}f
( a,
h,
9,
l
);
h,
b,
f,
m
9,
f,
c,
n
l,
m,
n,
d
moreover
\ = ad — P + nh — mg + 1, p = be — / 2 + nli — mg + 1,
p — bn— mf + dli —ml , a =fg — ch + gl — na ,
v = dg — nl +nf — cm ,
and A is the determinant
( a , h ,
h , b
9 > /“I,
l — 1, m ,
t = hf— bg + ma — Ih
9 >1+1)
f+1, m
c , n
n , d
56—2