318 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
Art. 15. If the proposed matrix ||a|| be not square, but of the type nX{n-\-m), let 
||a|j = ]|yj| X ||«'||, where ||a'|| is a prime matrix of the same type as ||a||, and ||v„|l a square 
matrix, of which the determinant is Vm the greatest divisor of ||«||. Then if ||v„|| be 
expressed in the form 
N X 
V?t Vw— 1 
Vra— 1 V«— 2 
Vo 
X 
and if, for brevity, we write ||Vj| for ||«^|| x |a'||, we obtain for ||a|| the expression 
X||V|| 
ix 
V« V«— 1 
— 9 9 . 
Vi 
<1 
1 
<1 
1 
to 
Vo 
(79.) 
The numbers V)» V«-n • • •? which are the greatest common divisors of the minors of 
jVj, are also by the theorem of art. 12, the greatest common divisors of the minors 
of ||a||. We see therefore that — ^ is always divisible by in the case of an oblong 
Vs— I Vs— 2 
as well as a square matrix. 
Art. 16. To show still more clearly the nature of the quotients we add the 
Vs-l 
following proposition : — 
“ If in any rectangular matrix we divide each minor determinant of order s by the 
greatest common di\isor of its own first minors, the greatest common divisor of all the 
quotients thus obtained is 
Vs-l 
By this proposition — ^ is itself defined as a greatest common divisor, instead of being 
Vs— I 
defined as the quotient of one greatest common divisor, divided by another. 
To establish its truth we may first consider the quotient in any rectangular 
matrix ||A|| of the type nx{')n-\-n). Let oj denote the greatest common divisor of the 
quotients obtained by di\iding each determinant of 1|A|| by the greatest common divisor 
of the first minors of that determinant : we have then to show that 
v» 
V«— 1 
= 00. 
Since the greatest common divisor of any vertical column of minors in ||A|| is not 
altered by premultiplication with a unit-matrix, it is evident that a> as well as 
will remain unchanged by that operation. If, therefore. 
l=Hx 
Vn Vn— 1 
9 9 . 
Vi 
Vrt— 1 Vn— 2 
Vo 
XllVll 
(79.) 
where ll-yjl is a unit, and ||V| a prime matrix, we may consider instead of ||A||, the simpler 
matrix 
V« V«— 1 
9 9 . . . 
V«— 1 Vn— 2 Vo 
x||V||. 
(80.) 
Let ll^gll, . . . &c. be the difierent square matrices of ||V|| ; 0^, . . . their determi- 
