816 
ME. H. J. S. SMITH ON SYSTEMS OP LINEAE 
ill which 
1 1 
— ) J., -L, • • • 
Vn— 1 
denotes a matrix of the type nXn. It follows from (73.) 
that v„_, is the determinant of ||^|| ; let that matrix be reduced by premultiplication 
u'ith a unit-matrix ; and let 
IHI=IMIx||v»-4 0 ^-) 
where ||«;j| is the reducing unit, and ||v«-i|i reduced matrix, 
^1, 
^’l, 2? 
3 • 
0, 
^2 ? 
^2,3 • 
• o 
0 , 
(^3 • 
0, 
0 , 
0 .. 
(75.) 
so that (73.) assumes the form 
H=Hxl|v«-,llx 
Vn — 1 
, 1 , 1 ,... 
X w 
(76.) 
It may be proved that in (75.) = r, 2=0, r,_3=0 .. ri_„=0. For since the matrix 
, 1, 1.. is derived from ||«|| by multiplication with unit-matrices, v«-i is the 
X 
v« 
,Vh-i 
greatest common divisor of the first minors of ||Vn-i||X 
v« 
iV»-i 
-, 1, 1, .. 
Therefore Vn-i 
divides ■ - X which is one of those minors ; but also y„_, =(«., x /M '2 • • X ; ?• 
^^2XiM'3X .. X^/.'„=V«-i, and the product ||v„-)||X 
Vn— 
1 , 1 ,- 
assumes the form 
Vn 
V«-i’ 
^1, 25 
^1,3? • 
• • n.» 
0 , 
[^■'2 5 
^’ 2 , 3 ? • 
• • ^2,« 
0 , 
0 , 
H-'s •> • 
• . rs,n 
0 , 
0 , 
0 
One of the minors of this matrix is • • • X («•«, which cannot be divisible by v«-i 
or (jj^X (Jt^sX . • -X unless ^ is a multiple of but ^ because llv»-i|| is reduced, 
therefore 2=0. Similarly, it may successively be shown that 3=0 . . . „=0. Now 
if the matrix 
^^2 5 ^2, 3? • • • ^2, n 
0 i '^3, n 
0,0 , . . . 
which is of the type (^^ — l)x (w— 1), be reduced to the form 
( 77 .) 
