IKDETEEMEs’ATE EQUATIONS AND CONOEUENCES. 
309 
1 , 
0 , 
0 , . 
. 0 
,0 ,. 
.0 
0 , 
1 , 
0 , . 
. 0 
,0 ,. 
.0 
0 , 
0 , 
1 ,. 
. 0 
,0 ,. 
.0 
0 , 
0 , 
0 , . 
. Ui ^ 
1 i 1 , 2 ? 
0 , 
0 , 
0 , . 
. 2 ^ 2 ^ 
1 ? '^^ 2,21 
• • « 
0 , 
0 , 
0 , . 
• Un , 
1 ? 2 ? 
• • n 
j|A| denoting a matrix of the type ny^m oi which the constituents may be any numbers 
whatever, and || 2 ;[| a unit-matrix of the t}"pe nyn. If for |jx|| we substitute the matrix 
it is readily seen that we may invert the order of the factors in the expression of H'yl ; so 
that, using an abbreriated notation, the signification of which is evident, we may write 
either 
or 
V = 
1, 0| 
y, ll 
X 
1,0 
0, u ’ 
1, 0, 
0, u, 
jx 
O 1-H 
Substituting the latter expression of ||y|l in the equation 
we immediately infer 
Every matrix satisfying the equation 
V X 
kl!=Mx||y+2/AC|l. 
c 
X 
= 1 is therefore comprised in the formula (55.) ; 
and since it is erident, conversely, that every matrix comprised in (55.) satisfies the 
equation, that formula contains the complete solution of the question. 
A particular solution of the problem (which may be taken for ||yl|) can be obtained 
as follows : — Complete the matrix |jC|| by any n horizontal rows of constituents which do 
not cause the determinant of the resulting matrix to vanish. From this matrix a prime 
{i. e. a unit) matrix of the same type is to be deduced by the method of art. 3, a reduc- 
tion which can always be effected without changing the prime matrix ||C||. 
Art. 11. The consideration of sets of fundamental solutions of linear systems is also 
of use in the theory of indeterminate systems containing terms not affected by any inde- 
terminate. Let 
Ai, 0 + ^ 1 , 1 + 2^2+ • • • n + ?« 
z=l, 2, 3, ... w ^ 
represent such a system ; its general solution will assume the form 
+ — 7 t % 
y-'e ^k, e 1 
^=1, 2, 3, . . . n-\-7nj 
MDCCCLXI. 2 U 
(56.) 
( 57 .) 
