Determinants and Their Applications in Mathematical Physics

36 3. Intermediate Determinant Theory

=

∑

r

frA

rj

δri+

∑

s

gsA

is

δsj

=fiA

ij

+gjA

ij

which proves (D). The proof of (C) is similar but simpler.

Exercises

Prove that

1.

n

∑

r=1

n

∑

s=1

[r

k

−(r−1)

k

+s

k

−(s−1)

k

]arsA

rs

=2n

k

.

2.a

′

ij

=−

n

∑

r=1

n

∑

s=1

aisarj(A

rs

)

′

.

3.

n

∑

r=1

n

∑

s=1

(fr+gs)aisarjA

rs

=(fi+gj)aij.

Note that (2) and (3) can be obtained formally from (B) and (D), respec-

tively, by interchanging the symbolsaandAand either raising or lowering

all their parameters.

3.5 The Adjoint Determinant...................

3.5.1 Definition........................

The adjoint of a matrixA=[aij]nis denoted by adjAand is defined by

adjA=[Aji]n.

The adjoint or adjugate or a determinantA=|aij|n= detAis denoted by

adjAand is defined by

adjA=|Aji|n=|Aij|n

= det(adjA). (3.5.1)

3.5.2 The Cauchy Identity

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

Theorem.

adjA=A

n− 1

.

The proof is similar to that of the matrix relation

AadjA=AI.