308 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
of an equation of the form 
C,a’i+C2^'2+ +C,i+rT„+, = l 
can be exhibited in the determinantal form (50.), is occasionally useful. 
The preceding problem is a particular case of the following more general enunciation ; — 
Given a prime matrix ||C|| of the type to find all the matrices la’ll of the 
type nX(ni+7«) which satisfy the equation 
X 
(64.) 
I^et Ijyjl be a matrix which satisfies (54.), let the numbers represent absolute inde- 
terminates, and ||«1| any unit-matrix ; the complete solution of the problem is contained 
in the formula 
1!'^||=||m|| X |i7+2f.*C||, (^b.) 
where C|| represents the matrix, 
+ ^0=m r-i 
= 
f=l, 2, 3, ... w 
j=l, 2, 3, . . . n+m. 
For if Ija’jl be a matrix satisfying the equation (54.), we have 
and consequently 
c 
— 1 — 
c 
X 
7 
= N X 
|?'i| denoting a unit of the type X {m-{-n). But because the first m horizontal rows 
are identical, it is evident that 
in 
C'l 
and 
lC| 
XI 
,7 
_ ^=l, 2, 3, ... m 
' T O 9 I 
^=1, 2, 3, . . . m-\rn. 
except when ^==/, in which case 
'^1, l~^2, 2— • • • '^m, m — !• 
The unit-matiix ||i;|| therefore arises from the composition of two unit-matrices of the 
forms 
1 ,0 , 
0 ,. 
. 0 
o 
o 
. 0 
0 ,1 , 
0 
. 0 
9 
0, 0,. 
. 0 
. o 
o 
1 
. 0 
0, 0,. 
. 0 
^1, 1? ^1, 2? 
^1,3? • 
• ^1, 
1, 0, . 
. 0 
^2, n ^2, 2i 
^2, 3? • 
. 72, 
W? 
0, 1, . 
. 0 
2? 
3? • 
. 
7/»9 
o 
o 
. 1 
