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