INDETEEMIXATE EQUATIONS AND CONOEUEXCES. 
311 
be applied to any independent system, equivalent to the proposed system, and deduced 
linearly from it. 
If Tve represent by the greatest divisor of the matrix, deduced from the matrix of 
(59.) by omitting from it the column A, j, Ag, t, . . . A„ j, we may enunciate the follow- 
ing proposition : — 
“In every solution of the system (59.), the value of .ry is divisible by yy; and, con- 
versely, a solution of that system can always be assigned in which Xy. shall have any 
T) 
given value dhisible by -yy.” 
It will be seen that the solution of (56.) depends, first, on the solution of (59.), and, 
secondly, on that of the indeterminate equation 
^1, 0 0 I, 0 + l — 1- 
If we represent the values of the indeterminates in this equation as the determinants of 
the matrix 
2 = 1, 2, 3, . . . m 
j=l, 2, .. . w+I 
(see art. 10), we may express the most general values of the indeterminates which 
satisfy (56.) in the determinantal form 
^ 7/1 + 1 , k 
^k= 
Vi, i ^ 1 , 05 
7\,2 +|“'l ^2,05 • • 
72. 1 +|“'2 05 
72, 2 "F f^2 ^2, 05 • • 
7m, 1 “1“ ^ 1 , 05 
/m, 2 1 ^2, 0’ * ' 
Art. 12. We shall now indicate an important transformation of which any square 
matiix of integral numbers is susceptible. We begin with the following theorem: — 
“ If a given rectangular matrix be premultiplied by a unit matrix, the greatest com- 
mon divisor of any vertical column of minor determinants is the same in the resulting 
as in the given matrix.” 
For it is evident that any minor, either in the given or in the resulting matrix, is an 
integral and linear function of the minors formed from the same vertical columns in the 
other matrix. 
Similarly, it may be shown that 
“When a square matrix is post-multiplied by any prime rectangular matrix, the 
gi’eatest common divisor of any horizontal row of minors is the same in the resulting- 
rectangular matrix as in the given square matrix,” 
For if 
n X {n-\-in) 
nXn 
[ 
1 V 
A 
B 
A 
C 
2 u 2 
