Page 39 - Determinants and Their Applications in Mathematical Physics

24 3. Intermediate Determinant Theory

etc. In simple algebraic relations such as Cramer’s formula, the advantage
of using scaled rather than simple cofactors is usually negligible. The Jacobi
identity (Section 3.6) can be expressed in terms of unscaled or scaled cofac-
tors, but the scaled form is simpler. In differential relations, the advantage
can be considerable. For example, the sum formula
n
(n)
a ij A
kj = A n δ ki
j=1
when differentiated gives rise to three terms:
n
(n) (n)

a A + a ij (A ) = A δ ki .

ij kj kj n
j=1
When the cofactor is scaled, the sum formula becomes
n

a ij A kj (3.2.13)
n = δ ki
j=1
which is only slightly simpler than the original, but when it is differentiated,
it gives rise to only two terms:
n

kj
a A kj + a ij (A ) =0. (3.2.14)
ij n n
j=1
The advantage of using scaled rather than unscaled or simple cofactors will
be fully appreciated in the solution of differential equations (Chapter 6).
Referring to the partial derivative formulas in (2.3.10) and Section 3.2.3,
∂A ip ∂
= A ip
A
∂a jq
∂a jq

1 ∂A ip ∂A
= A
A 2 ∂a jq − A ip ∂a jq
= 1
A 2 AA ij,pq − A ip A jq
= A ij,pq − A A . (3.2.15)
jq
ip
Hence,

∂
A jq + ∂a jq A ip = A ij,pq . (3.2.16)
Similarly,

∂
A kr + ∂a kr A ij,pq = A ijk,pqr . (3.2.17)
The expressions in brackets can be regarded as operators which, when
applied to a scaled cofactor, yield another scaled cofactor. Formula (3.2.15)