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

∑

j=1

aijA

(n)

kj

=Anδki

when differentiated gives rise to three terms:

n

∑

j=1

[

a

′

ijA

(n)

kj

+aij(A

(n)

kj

)

′

]

=A

′

nδki.

When the cofactor is scaled, the sum formula becomes

n

∑

j=1

aijA

kj

n

=δki (3.2.13)

which is only slightly simpler than the original, but when it is differentiated,

it gives rise to only two terms:

n

∑

j=1

[

a

′

ij

A

kj

n

+aij(A

kj

n

)

′

]

=0. (3.2.14)

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

∂ajq

=

∂

∂ajq

(

Aip

A

)

=

1

A

2

[

A

∂A

ip

∂ajq

−Aip

∂A

∂ajq

]

=

1

A

2

[

AAij,pq−AipAjq

]

=A

ij,pq

−A

ip

A

jq

. (3.2.15)

Hence,

(

A

jq

+

∂

∂ajq

)

A

ip

=A

ij,pq

. (3.2.16)

Similarly,

(

A

kr

+

∂

∂akr

)

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)