Determinants and Their Applications in Mathematical Physics

30 3. Intermediate Determinant Theory

Now, interchange the dummies wherever necessary in order thatp<

q<rin all sums. The result is

A=

∑

p<q<r

[

aipajqakr−aipajrakq+aiqajrakp

−aiqajpakr+airajpakq−airajqakp

]

Aijk,pqr

=

∑

p<q<r

∣ ∣ ∣ ∣ ∣ ∣

aip aiq air

ajp ajq ajr

akp akq akr

∣ ∣ ∣ ∣ ∣ ∣

Aijk,pqr

=

∑

p<q<r

Nijk,pqrAijk,pqr, i<j<k,

which proves the Laplace expansion formula from rowsi,j, andk.

3.3.3 Determinants Containing Blocks of Zero Elements

LetP,Q,R,S, andOdenote matrices of ordern, whereOis null and let

A 2 n=

∣

∣

∣

∣

PQ

RS

∣

∣

∣

∣

2 n

.

The Laplace expansion ofA 2 ntaking minors from the first or lastnrows or

the first or lastncolumns consists, in general, of the sum of

(

2 n

n

)

nonzero

products. If one of the submatrices is null, all but one of the products are

zero.

Lemma.

a.

∣

∣

∣

∣

PQ

OS

∣

∣

∣

∣

2 n

=PS,

b.

∣

∣

∣

∣

OQ

RS

∣

∣

∣

∣

2 n

=(−1)

n

QR

Proof. The only nonzero term in the Laplace expansion of the first

determinant is

N 12 ...n;12...nA 12 ...n;12...n.

The retainer minor is signless and equal toP. The sign of the cofactor is

(−1)
k
, wherekis the sum of the row and column parameters.

k=2

n

∑

r=1

r=n(n+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

Nn+1,n+2,..., 2 n;12...nAn+1,n+2,..., 2 n;12...n.