Page 57 - Determinants and Their Applications in Mathematical Physics

42 3. Intermediate Determinant Theory

= − δ rp δ sq A A rj
is
r s
= −A A pj
iq
1
= − [A iq A pj ]. (3.6.7)
A 2
Hence,

A ij A iq = AA ip,jq , (3.6.8)
A pj A pq
which, when the parameter n is 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 ipu,jqv ,

A uj A uq A uv
where the cofactors are scaled.
Proof. From (3.2.4) and Theorem 3.4,
2
∂ A = A pu,qv
∂a pq ∂a uv
= AA pu,qv

A pq A
pv
= A . (3.6.9)
A uq A
uv
Hence, referring to (3.6.7) and the formula for the derivative of a
determinant (Section 2.3.7),
3
∂ A
∂a ij ∂a pq ∂a uv
pq pv
∂A A A pq ∂A
A pq A pv
∂A pv ∂a ij
= uq ∂a ij
A uq A uv + A ∂A uv + A uq ∂A uv
A A

∂a ij ∂a ij
∂a ij

A pq A A pj A A pq A
pv pv pj
− AA iq − AA iv
A uq A uv A uj A uv A uq A uj
= A ij

1
= A pq A pv A pj A pv
A 2 A ij A uq A uv − A iq A uj A uv

A pj A pq

+ A iv
A uq
A uj

1 A ij A iq A iv
= . (3.6.10)
A 2 A pj A pq A pv
A uj A uq A uv