INDETEEMmATE EQUATIONS AND CONGEUBNCES. 297 
which e\idently implies that 
|A,|=|Ao|.; 
whence, by the theorem of the last article, 
lAil]— 1|®|| X ||Aq|[, 
l|a| denoting a unit-matrix. Combining (9.) and (11.), we find 
ll^ill X ||Cs|| X ||A()||=[j^oj| X ||Ao||, 
whence, by the same theorem, it follows that 
IMxMHM’ 
or, which is the same thing. 
( 10 .) 
( 11 .) 
(12.) 
(13.) 
|!>?;.l|=P„||x|Hr‘, (14.) 
denoting the matrix reciprocal to j|a||. The complete solution of equation (4.) is 
therefore contained in the formulae 
||Ai=t|« iixM I 
||K||=]|K.|lx||<» ||-'J 
jjajj denoting an arbitraiy unit-matrix of the type nXn, and ||Ao||, ||^o|| being any two 
matrices that satisfy the equation. 
In this, as in the preceding article, we have for simplicity excluded the case in which 
«^=0, and the matrices j|Kj| and ||A|| are squares. But it is readily seen that no exception 
is presented by this particular case. 
Art. 4. Let 
A/j 1 ^i-j-A,- 2 ••• “1“A,- „ + ^'n+m — b,| 
? = 1, 2, 3, ...w J 
. (16. 
represent a system of indeterminate equations of which the matrix is ||A||. We shall 
suppose that the determinants of ||A|| are not all equal to zero, i. e. that the system is 
independent ; so that its index of indeterminateness (or the excess of the number of 
indeterminates above the number of really independent equations) is m. If we take r 
solutions of the system, for example the solutions 
n 25 3 ■ • • ^s, n + m5l 
s=l, 2, 3, ...r 
J 
(17.) 
it is evident that if r>m, the determinants of the matrix |i.r|j are all equal to zero. If 
w, and if the determinants of the matrix j|a’|| be not all equal to zero, the solutions 
(17.) are said to form a set of r independent solutions', if r=m, they form a complete 
set of independent solutions. A set of relatively prime solutions is an independent set of 
which the matrix is prime ; a complete set of relatively prime solutions may be called, 
for a reason which will presently appear, a fundamental set of solutions. It is always 
possible, in an infinite number of ways, to assign complete sets of independent solutions 
of a system of equations of the form (16.). Among the methods by which this may be 
accomplished, we shall select one which depends on the following principle : — 
