IXDETEEMINATE EQUATIONS AND CONGRUENCES. 
301 
the relation in question is expressed by the formula 
kx\A\=Kx\ci 
(25.) 
where it is to be remembered that the types of the matrices ||A|| and ||fl;|l are comple- 
mentary; so that, as has been already observed (see art. 1), there is an ambiguity of 
sign in the equation (25.). 
To obtain its demonstration, let Q and q denote the sums of the squares of the deter- 
A 
minants of ||A|| and ||a|[ respectively, and consider the determinant 
is certainly not zero, for multiplying it by itself, we find 
A^ 
Let, then, 
This determinant 
(26.) 
be multiplied by any determinant of ||A|| ; for example, by 2 + Ai_j, Ag , 
. . . A„ „. Observing that 2+A,^ „ Ao^a, • • • A„_„ may assume the form 
1 J -^2, 1 ? • • 
. A„, 1 , 0, 0, . . . 
.. . 0 
-^I, 2 5 -^"^2, 2 ? • • 
• 2 ,0,0,... 
... 0 
-^1, re ? -^2, re : • • 
• Are_ „ ,0,0,... 
. .. 0 
A A 
■^^1, n + 1 5 "^^2, n + 1 ^ • * 
A 10 
• -^-*- 0 , re + 1 1 -*■; • • • 
. .. 0 
A A 
-^I, re + 2 1 ■‘^■2, re + 2 ? • • 
A D 1 
• n + 2 ? '^9 -*-9 • • • 
...0 
'"^1, n + m^ -^2, n + m? • * 
• n-bm9 ^9 ^9 * * * 
.. . 1 
we obtain the equation 
A 
a 
condition that 
X n -^2,2? • • • -^n, re Q X ,, „ + ••• ^^m,n + rn^ * • (^G) 
in which we may permute the second set of indices in any manner consistent with the 
should not change its sign ; so that we may write 
x|A|=QxH, (28.) 
the coiTespondence of the determinants in [A] and \a\ being fixed by the matrix ^ . 
€C 
The equation (25.) is an immediate consequence of this result; and if in that equation 
we suppose the correspondence of the determinants to be still fixed by the matrix 
we shall have to write 
or 
according as 
kx\A\= Kx\a\, 
^XlA|=— Kx|4 
is a positive or negative number. 
2 T 
ilDCCCLXI. 
