Determinants and Their Applications in Mathematical Physics

282 6. Applications of Determinants in Mathematical Physics

6.9.2 Three Determinants..................

The determinantAand its cofactors are closely related to the Matsuno

determinantEand its cofactor (Section 5.4)

A=KnE,

2 crArs=KnErs,

4 crcsArs,rs=KnErs,rs,

where

Kn=2

n

n

∏

r=1

cr. (6.9.4)

The proofs are elementary. It has been proved that

n

∑

r=1

Err=

n

∑

r=1

n

∑

s=1

Ers,

n

∑

r=1

n

∑

s=1

Ers,rs=− 2

n

∑

r=1

n

∑

s− 1

†

csErs.

It follows that

n

∑

r=1

crArr=

n

∑

r=1

n

∑

s=1

crArs (6.9.5)

n

∑

r=1

n

∑

s=1

crcsArs,rs=−

n

∑

r=1

n

∑

s=1

†

crcsArs. (6.9.6)

Define the determinantBas follows:

B=|bij|n,

where

bij=







aij− 1

ci+cj

ci−cj

,j=i

ωθi,j=i (ω

2

=−1).

(6.9.7)

It may be verified that, for all values ofiandj,

bji=−bij,j=i,

bij−1=−a

∗

ji

,

a

∗

ij

−1=−bji. (6.9.8)

Whenj=i,a
∗
ij
=aij, etc.

Notes on bordered determinants are given in Section 3.7. LetPdenote

the determinant of order (n+ 2) obtained by borderingAby two rows and