814 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
/. e. that only one post-multiplying unit can be assigned by which a given matrix can be 
reduced to the form (62.). 
If instead of reducing the given matrix ||a|| by post-multiplication we employ a pre- 
multiplying unit, we obtain the following theorem : — 
“ Every matrix of the type ny^n and of determinant d is equivalent (by pre-multipli- 
cation) to one, and only one of the matrices included in the formula (62.), in which 
. . are positive, (Jb .^ . . p„=^, and ,, satisfies the inequality 
( 66 .) 
Art. 14*. The transformation to which Ave have referred in art. 12 is obtained by 
employing simultaneously a pre-multiplying and a post-multiplying unit-matrix. It is 
expressed by the equation 
\a\ 
Vw— 1 
1 5 — ? 
1 Vn — 2 
(67.) 
in which |[ff|| is a given square matrix of the type w X ]|a|| and ||/3j| are unit-matrices, and 
V«- 1 , V„- 2 , • • • Yi, Vo are the determinant and greatest common divisors of the 
minor determinants of |jaj|, so that, in particular, v« is the determinant of ||a||, Vn-i liie 
greatest common divisor of its minor determinants of order n — 1, Vi Ih© greatest 
common divisor of its constituents, and Vo = l. -i’he units l|a|| and ||j3|| are not abso- 
lutely determined, but admit, when ^^>l, of an infinite number of different values. 
If « = 1, it is evident that the formula (67.) is verified; for we have the identical equa- 
tion ||«jjz=||l||x 
Vo 
X 
It is therefore sufficient to show that, if the transformation 
indicated in the formula can be effected for matrices of the type {n—l)x{n — 1), it 
can also be effected for matrices of the type n X n. The demonstration depends on an 
elementary principle, which it is worth while to enunciate separately. 
•‘If 
Ui 1^1 2^2 “h • • • n + + 
i—1, 2, 3, ... w J 
( 68 .) 
denote a system of linear functions of n-\-m indeterminates, (m>0), and if the consti- 
tuents of the matrix {jA|| do not admit of any common divisor, it is always possible to 
assign integral values to . . . x„+^^ which shall render Ui, Ug, . . . U„ relativelj 
prime.” 
For, in the first place, we can obtain values for Ui, Ug, . . . U„, which shall not have 
any common divisor with a given number M. Let q, r ... he the different prime 
divisors of M ; one at least of the constituents of ||A||, for example A;_y, is prime top. 
Attributing to Xj a value prime to p., and values divisible by p to the remaining indeter- 
minates, we shall obtain for Uj a value which is certainly prime to p. Similarly, by sub- 
jecting the indeterminates to proper congruential conditions with respect to the modules 
q, r. . . ., we can render one, at least, of the functions U prime to q, one prime to r, and 
* [This article has been in great part rewritten since the paper was read. The demonstration is not 
essentially changed, but is presented in what seems to be a simpler form. — Sept. 1861, IT. J. S. S.] 
