Determinants and Their Applications in Mathematical Physics

42 3. Intermediate Determinant Theory

=−

∑

r

∑

s

δrpδsqA

is

A

rj

=−A

iq

A

pj

=−

1

A

2

[AiqApj]. (3.6.7)

Hence,

∣

∣

∣

∣

Aij Aiq

Apj Apq

∣

∣

∣

∣

=AAip,jq, (3.6.8)

which, when the parameternis restored, is equivalent to (3.6.4). The

formula given in the theorem follows by scaling the cofactors.

Theorem 3.5.

∣ ∣ ∣ ∣ ∣ ∣

A

ij

A

iq

A

iv

A

pj

A

pq

A

pv

A

uj

A

uq

A

uv

∣ ∣ ∣ ∣ ∣ ∣

=A

ipu,jqv

,

where the cofactors are scaled.

Proof. From (3.2.4) and Theorem 3.4,

∂

2

A

∂apq∂auv

=Apu,qv

=AA

pu,qv

=A

∣

∣

∣

∣

A

pq

A

pv

A

uq

A

uv

∣

∣

∣

∣

. (3.6.9)

Hence, referring to (3.6.7) and the formula for the derivative of a

determinant (Section 2.3.7),

∂

3

A

∂aij∂apq∂auv

=

∂A

∂aij

∣

∣

∣

∣

A

pq

A

pv

A

uq

A

uv

∣

∣

∣

∣

+A

∣

∣

∣

∣

∣

∂A

pq

∂aij

A

pv

∂A

uq

∂aij

A

uv

∣

∣

∣

∣

∣

+A

∣

∣

∣

∣

∣

A

pq ∂A

pv

∂aij

A

uq ∂A

uv

∂aij

∣

∣

∣

∣

∣

= Aij

∣

∣

∣

∣

A

pq

A

pv

A

uq

A

uv

∣

∣

∣

∣

−AA

iq

∣

∣

∣

∣

A

pj

A

pv

A

uj

A

uv

∣

∣

∣

∣

−AA

iv

∣

∣

∣

∣

A

pq

A

pj

A

uq

A

uj

∣

∣

∣

∣

=

1

A

2

[

Aij

∣

∣

∣

∣

Apq Apv

Auq Auv

∣

∣

∣

∣

−Aiq

∣

∣

∣

∣

Apj Apv

Auj Auv

∣

∣

∣

∣

+Aiv

∣

∣

∣

∣

Apj Apq

Auj Auq

∣

∣

∣

∣

]

=

1

A

2

∣ ∣ ∣ ∣ ∣ ∣

Aij Aiq Aiv

Apj Apq Apv

Auj Auq Auv

∣ ∣ ∣ ∣ ∣ ∣

. (3.6.10)