296 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
the same type as ||K||, and in which the constituents of ||A|| and ||/?:|| are the unknown 
numbers. 
We shall first obtain a particular solution of this equation, and then show how from 
any particular solution the complete solution may be deduced. 
We may suppose that the constituents of any horizontal row of |1K|| admit of no 
common divisor but unity ; for if ^ 2 , ... be the greatest common divisors of the 
constituents of the horizontal rows of ||K||, Ave find 
|iK|l=p.,^2,^3, ..^JxllK'll, (5.) 
||K'|| denoting a matrix the constituents of which are derived from those of ||K|1 by the 
relation 
K' — -K • 
S § s ’ 
(6.) 
so that the solution of equation (4.) depends on the solution of a similar equation for 
the matrix |jK'|[, in which the constituents of each horizontal row are relatively prime. 
Let then the matrix + 
K 
, e. the matrix 
1^1, 1: ^1, 25 • • 
• • Kl, n + m 
^2, 15 1^2^ 25 * • 
• • -^2, n+m 
15 25 • • 
• • n+m 
K 
be a prime matrix, but let the matrix 
Determine cu^, — c^r by the system of congruences 
<^ 2 + ••• mod. 
[1 </•<%], 
admit of a greatest divisor (A. 
i=l, 2, 3, . . . n-\-m 
J' ■ 
0-) 
(Avhich, as Ave have just seen, is ahvays resoluble), and in ||K|| replace the constituents 
Kr+i,,- by the numbers 
we thus deduce from |jK| ano.ther matrix ||K"|| connected with it by the relation 
|K|=|M, X |K"|, and such that the matrix of its first r+1 horizontal rows is prime. By 
proceeding in this manner, Ave shall at last obtain a prime matrix [[Ao||, which satisfies 
the equation |K|=A;xiAo|; Ave may then, by the method of the last article, determine a 
square matrix |ji?’o|j satisfying the equation 
i|Ki|=W|x||A„||, (8.) 
and thus obtain a particular solution of the proposed equation (4.). 
To deduce the general solution of that equation, let pi|| and |A,j] be any two matrices 
satisfying it. We have therefore the equality 
||/r.||x||A,|l=lWxl|A„lI, 
(9:) 
