Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 31

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

(−1)

k

, where

k=

n

∑

r=1

(n+2r)=2n

2

+n.

Hence, the cofactor is equal to (−1)

n

Q. Part (b) of the lemma follows. 

Similar arguments can be applied to more general determinants. LetXpq,

Ypq,Zpq, andOpqdenote matrices withprows andqcolumns, whereOpq

is null and let

An=

∣

∣

∣

∣

Xpq Yps

Orq Zrs

∣

∣

∣

∣

n

, (3.3.14)

wherep+r=q+s=n. The restrictionp≥q, which impliesr≤s, can

be imposed without loss of generality. IfAnis expanded by the Laplace

method taking minors from the firstqcolumns or the lastrrows, some

of the minors are zero. LetUmandVmdenote determinants of orderm.

Then,Anhas the following properties:

a.Ifr+q>n, thenAn=0.

b.Ifr+q=n, thenp+s=n,q=p,s=r, andAn=XppZrr.

c. Ifr+q<n, then, in general,

An= sum of

(

p

q

)

nonzero products each of the formUqVs

= sum of

(

s

r

)

nonzero products each of the formUrVr.

Property (a) is applied in the following examples.

Example 3.2. Ifr+s=n, then

U 2 n=

∣

∣

∣

∣

En, 2 r Fns Ons

En, 2 r Ons Fns

∣

∣

∣

∣

2 n

=0.

Proof. It is clearly possible to performnrow operations in a single step

andscolumn operations in a single step. RegardU 2 nas having two “rows”

and three “columns” and perform the operations

R

′

1

=R 1 −R 2 ,

C

′

2 =C 2 +C 3.

The result is

U 2 n=

∣

∣

∣

∣

On, 2 r Fns −Fns

En, 2 r Ons Fns

∣

∣

∣

∣

2 n

=

∣

∣

∣

∣

On, 2 r Ons −Fns

En, 2 r Fns Fns

∣

∣

∣

∣

2 n

=0

since the last determinant contains ann×(2r+s) block of zero elements

andn+2r+s> 2 n.