The So-called Vahleii Relations between the Minors of a Matrix. 697 
discussion is to take every primary minor of the given array in succession 
for the multiplicand, and the ad jugate of the first of them for the constant 
multiplier and make application of the theorem. If the array consist of 
h rows and m columns, there are thus (m),^ multiplications to be performed, 
and in regard to these he reasons as follows. First of all the multiplicand 
may have all its columns in common with the first minor, in which case the 
outcome is manifestly nugatory. Or it may have h— \ columns in common, 
when there is a like outcome, as the right-hand member reduces at once to 
exactly the same form as the left-hand member. The number of such cases 
is k(m—k), there being one for every way in which we can replace a column 
of the multiplicand by an outside column. All the other cases, 
{m)k — 1 —Jc{m — Jc) 
in number, are real relations between the primary minors of the array. Not 
only so, but it is afiirmed that all these latter relations are mutually 
independent, the reason being that each one involves a minor, the multi- 
plicand, which necessarily does not appear in any of the others. 
(5) All this needs to be received with some caution and scrutiny. As a 
matter of fact, in the case of every one of the 
multiplications viewed by Vahlen as effective, the — 2)'^ power of the first 
of the primary minors can be removed from both sides, leaving on the left- 
hand the product of two minors, and on the right-hand an aggregate of 
products of pairs of other minors — leaving, that is to say, what Sylvester's 
theorem would have given at once. 
(6) As the matter is of some moment, let us illustrate by the case of a 
4-by-9 array, say the array 
aj . . . . h-^^ i-^ 
^2 h.) d.2 . . . . h^ ^2 
a^h^ d^ . . . . h^ i^ 
Here the number of minors is 126, and of the 126 multiplications 21 are 
properly called nugatory ; for example, the columnwise multiplication of 
\a^\c.^h^\ by IA1B2C3DJ 
gives 
\a^2'^?,^± 
'h^2H'^i\ l^l^2^3^4l l^l^2^3^4| l«l<^2^3^^4l 
