Page 46 - Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 31

The retainer minor is signless and equal to R. The sign of the cofactor is
(−1) , where
k
n
2
k = (n +2r)=2n + n.
r=1
Hence, the cofactor is equal to (−1) Q. Part (b) of the lemma follows.
n
Similar arguments can be applied to more general determinants. Let X pq ,

Y pq , Z pq , and O pq denote matrices with p rows and q columns, where O pq
is null and let

A n = X pq Y ps , (3.3.14)
O rq Z rs
n
where p + r = q + s = n. The restriction p ≥ q, which implies r ≤ s, can
be imposed without loss of generality. If A n is expanded by the Laplace
method taking minors from the first q columns or the last r rows, some
of the minors are zero. Let U m and V m denote determinants of order m.
Then, A n has the following properties:
a. If r + q> n, then A n =0.
b. If r + q = n, then p + s = n, q = p, s = r, and A n = X pp Z rr .
c. If r + q< n, then, in general,

A n = sum of p
q nonzero products each of the form U q V s

= sum of s nonzero products each of the form U r V r .
r
Property (a) is applied in the following examples.
Example 3.2. If r + s = n, then

U 2n = E n,2r F ns O ns =0.

E n,2r
O ns
F ns
2n
Proof. It is clearly possible to perform n row operations in a single step
and s column operations in a single step. Regard U 2n as having two “rows”
and three “columns” and perform the operations
R = R 1 − R 2 ,

1

C = C 2 + C 3 .
2
The result is

U 2n = O n,2r F ns −F ns

F ns
O ns
E n,2r
2n

= O n,2r O ns −F ns

F ns
F ns
E n,2r
2n
=0
since the last determinant contains an n × (2r + s) block of zero elements
and n +2r + s> 2n.