Page 45 - Determinants and Their Applications in Mathematical Physics

30 3. Intermediate Determinant Theory

Now, interchange the dummies wherever necessary in order that p<
q< r in all sums. The result is

A = a ip a jq a kr − a ip a jr a kq + a iq a jr a kp
p<q<r

− a iq a jp a kr + a ir a jp a kq − a ir a jq a kp A ijk,pqr

a ip a iq a ir
=
a jp a jq a jr A ijk,pqr
p<q<r a kp a kq a kr

= N ijk,pqr A ijk,pqr , i<j <k,
p<q<r
which proves the Laplace expansion formula from rows i, j, and k.

3.3.3 Determinants Containing Blocks of Zero Elements

Let P, Q, R, S, and O denote matrices of order n, where O is null and let

P Q
A 2n = .
R S
2n
The Laplace expansion of A 2n taking minors from the first or last n rows or

the first or last n columns consists, in general, of the sum of 2n nonzero
n
products. If one of the submatrices is null, all but one of the products are
zero.
Lemma.

P Q
a. = PS,
O S
2n

OQ

b. =(−1) QR
n
R S

2n
Proof. The only nonzero term in the Laplace expansion of the first
determinant is
N 12...n;12...n A 12...n;12...n .
The retainer minor is signless and equal to P. The sign of the cofactor is
(−1) , where k is the sum of the row and column parameters.
k
n

k =2 r = n(n +1),
r=1
which is even. Hence, the cofactor is equal to +S. Part (a) of the lemma
follows.
The only nonzero term in the Laplace expansion of the second
determinant is
N n+1,n+2,...,2n;12...n A n+1,n+2,...,2n;12...n .