Determinants and Their Applications in Mathematical Physics

34 3. Intermediate Determinant Theory

The result is:

AnBn=

∣

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

c 11 c 12 ... c 1 n

c 21 c 22 ... c 2 n

cn 1 cn 2 ... cnn

− 1 b 11 b 12 ... b 1 n

− 1 b 21 b 22 ... b 2 n

− 1 bn 1 bn 2 ... bnn

∣

∣

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 n

. (3.3.17)

The product formula follows by means of a Laplace expansion.cijis most

easily remembered as a scalar product:

cij=

[

ai 1 ai 2 ···ain

]

•







b 1 j

b 2 j

···

bnj





. (3.3.18)

LetRidenote theith row ofAnand letCjdenote thejth column of

Bn. Then,

cij=Ri•Cj.

Hence

AnBn=|Ri•Cj|n

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

R 1 • C 1 R 1 • C 2 ··· R 1 • Cn

R 2 • C 1 R 2 • C 2 ··· R 2 • Cn

······ ······ ··· ······

Rn•C 1 Rn•C 2 ··· Rn•Cn

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (3.3.19)

Exercise.IfAn=|aij|n,Bn=|bij|n, andCn=|cij|n, prove that

AnBnCn=|dij|n,

where

dij=

n

∑

r=1

n

∑

s=1

airbrscsj.

A similar formula is valid for the product of three matrices.

3.4 Double-Sum Relations for Scaled Cofactors.........

The following four double-sum relations are labeled (A)–(D) for easy refer-

ence in later sections, especially Chapter 6 on mathematical physics, where

they are applied several times. The first two are formulas for the derivatives