Page 51 - Determinants and Their Applications in Mathematical Physics

36 3. Intermediate Determinant Theory

= f r A δ ri + g s A δ sj
rj
is
r s
= f i A + g j A ij
ij
which proves (D). The proof of (C) is similar but simpler.

Exercises

Prove that
n n

1. [r − (r − 1) + s − (s − 1) ]a rs A rs =2n .
k
k
k
k
k
r=1 s=1
n n

rs
2. a = − a is a rj (A ) .
ij
r=1 s=1
n n

3. (f r + g s )a is a rj A rs =(f i + g j )a ij .
r=1 s=1
Note that (2) and (3) can be obtained formally from (B) and (D), respec-
tively, by interchanging the symbols a and A and either raising or lowering
all their parameters.
3.5 The Adjoint Determinant
3.5.1 Definition
The adjoint of a matrix A =[a ij ] n is denoted by adj A and is defined by

adj A =[A ji ] n .
The adjoint or adjugate or a determinant A = |a ij | n = det A is denoted by
adj A and is defined by

adj A = |A ji | n = |A ij | n
= det(adj A). (3.5.1)

3.5.2 The Cauchy Identity

The following theorem due to Cauchy is valid for all determinants.
Theorem.

adj A = A n−1 .
The proof is similar to that of the matrix relation
A adj A = AI.