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)
))