IXDETEEMINATE EQUATIO^^S AND CONGEDENCES. 319 
nants ; the greatest common divisor of those first minors in |j^j|| which do not contain 
the constituents of its uppermost row, so that ^ is integral ; lastly, let be the quotient 
obtained by di\iding the determinant 
Vn ^ V»— V 1 
Vtt-1 V«-2 Vo 
by the greatest common divisor of its first minors, so that a is the greatest common 
dmsor of . . . Now the greatest common divisor of the first minors of (81.) is 
evidently divisible by v«-n and divides v«-i X (because Vn-i'4'i is the greatest common 
divisor of one of its rows of minors). Consequently divides Vn-n and is divisible 
by Therefore is a common divisor of certain numbers respectively 
dividing the numbers . . ., viz. the numbers ; it is also (because ^i, . are 
relatively prime) the greatest common divisor of the numbers in which the same 
numbers are respectively contained ; ?. e. is the greatest common divisor of the 
Vn— 1 
numbers a;,, s/g • • • themselves, or 
Vn 
=Ct>. 
V n— 1 
By the aid of this particular case of the theorem the general proposition itself may be 
proved as follows : — 
If in any rectangular matrix of the type nX we propose to determine the 
greatest common divisor of the quotients obtained by dividing each minor determinant 
of order s, by the greatest common divisor of its own first minors, we may begin by 
selecting any s vertical columns [5<w], and forming the proper quotient for each deter- 
minant of order s, contained in this partial matrix of the type nxs. Let X, denote the 
greatest common divisor of these quotients ; then, as we have just seen, X,- is the greatest 
common divisor of all the determinants of the partial matrix, divided by the greatest 
common divisor of all its first minors. Hence (by art. 12) \ will remain unchanged 
when the given matrix is premultiplied by a unit-matrix. But is the greatest 
common divisor of all the divisors K 2 . . . con’esponding to every group of s vertical 
columns ; therefore Cl, is itself unchanged by premultiplication. Similarly, if a square 
matrix be post-multiplied by a rectangular prime matrix, it may be shown that is 
the same for the given square matrix, and for the resulting rectangular matrix. Hence 
if. as before. 
v» 
Vi 
v«-i’ 
' ' Vo 
X IIVII, 
Vs 
are the same for 
V» 
Vn— 1 
—11 
Vs-1 
V«-1 
Vn— 2 
Vo'l 
v« 
|V«-i’ 
■ ‘ Vo 
it is evident that and O, coincide ; therefore in any rectangular matrix 
Vs - 1 
Vs 
Vs-1 
2x2 
