Determinants and Their Applications in Mathematical Physics

3.4 Double-Sum Relations for Scaled Cofactors 35

A

′

and (A

ij

)

′

and the other two are identities:

A

′

A

= (logA)

′

=

n

∑

r=1

n

∑

s=1

a

′

rsA

rs

, (A)

(A

ij

)

′

=−

n

∑

r=1

n

∑

s=1

a

′

rsA

is

A

rj

, (B)

n

∑

r=1

n

∑

s=1

(fr+gs)arsA

rs

=

n

∑

r=1

(fr+gr), (C)

n

∑

r=1

n

∑

s=1

(fr+gs)arsA

is

A

rj

=(fi+gj)A

ij

. (D)

Proof. (A) follows immediately from the formula forA

′

in terms of un-

scaled cofactors in Section 2.3.7. The sum formula given in Section 2.3.4

can be expressed in the form

n

∑

s=1

arsA

is

=δri, (3.4.1)

which, when differentiated, gives rise to only two terms:

n

∑

s=1

a

′

rsA

is

=−

n

∑

s=1

ars(A

is

)

′

. (3.4.2)

Hence, beginning with the right side of (B),

n

∑

r=1

n

∑

s=1

a

′

rs

A

is

A

rj

=

∑

r

A

rj

∑

s

a

′

rs

A

is

=−

∑

r

A

rj

∑

s

ars(A

is

)

′

=−

∑

s

(A

is

)

′

∑

r

arsA

rj

=−

∑

s

(A

is

)

′

δsj

=−(A

ij

)

′

which proves (B).

∑

r

∑

s

(fr+gs)arsA

is

A

rj

=

∑

r

frA

rj

∑

s

arsA

is

+

∑

s

gsA

is

∑

r

arsA

rj