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.