Page 50 - Determinants and Their Applications in Mathematical Physics

3.4 Double-Sum Relations for Scaled Cofactors 35

A and (A ) and the other two are identities:

ij
A
n
n
= (log A) = a A , (A)

rs
A rs
r=1 s=1
n n

ij
(A ) = − a A A , (B)
is
rj
rs
r=1 s=1
n n n

(f r + g s )a rs A rs = (f r + g r ), (C)
r=1 s=1 r=1
n n

(f r + g s )a rs A A rj =(f i + g j )A . (D)
ij
is
r=1 s=1
Proof. (A) follows immediately from the formula for A 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

a rs A = δ ri , (3.4.1)
is
s=1
which, when differentiated, gives rise to only two terms:
n n

is is
a A = − a rs (A ) . (3.4.2)
rs
s=1 s=1
Hence, beginning with the right side of (B),
n n

a A A rj = A rj a A is

is
rs rs
r=1 s=1 r s

is
= − A rj a rs (A )
r s

is
= − (A ) a rs A rj
s r

is
= − (A ) δ sj
s
= −(A )
ij
which proves (B).

(f r + g s )a rs A A rj
is
r s

= f r A rj a rs A + g s A is a rs A rj
is
r s s r