Determinants and Their Applications in Mathematical Physics

3.5 The Adjoint Determinant 37

Proof.

AadjA=|aij|n|Aji|n

=|bij|n,

where, referring to Section 3.3.5 on the product of two determinants,

bij=

n

∑

r=1

airAjr

=δijA.

Hence,

|bij|n= diag|A A ... A|n

=A

n

The theorem follows immediately ifA=0.IfA= 0, then, applying (2.3.16)

with a change in notation,|Aij|n= 0, that is, adjA= 0. Hence, the Cauchy

identity is valid for allA.

3.5.3 An Identity Involving a Hybrid Determinant

LetAn=|aij|nandBn=|bij|n, and letHijdenote the hybrid determinant

formed by replacing thejth row ofAnby theith row ofBn. Then,

Hij=

n

∑

s=1

bisAjs. (3.5.2)

Theorem.

|aijxi+bij|n=An

∣

∣

∣

∣

δijxi+

Hij

An

∣

∣

∣

∣

n

,An=0.

In the determinant on the right, thexiappear only in the principal diagonal.

Proof. Applying the Cauchy identity in the form

|Aji|n=A

n− 1

n

and the formula for the product of two determinants (Section 1.4),

|aijxi+bij|nA

n− 1

n =|aijxi+bij|n|Aji|n

=|cij|n,

where

cij=

n

∑

s=1

(aisxi+bis)Ajs

=xi

n

∑

s=1

aisAjs+

n

∑

s=1

bisAjs

=δijAnxi+Hij.