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