Page 52 - Determinants and Their Applications in Mathematical Physics

3.5 The Adjoint Determinant 37
Proof.

A adj A = |a ij | n |A ji | n
= |b ij | n ,
where, referring to Section 3.3.5 on the product of two determinants,

n

b ij = a ir A jr
r=1
= δ ij A.
Hence,

|b ij | n = diag|A A ... A| n
= A .
n
The theorem follows immediately if A =0. If A = 0, then, applying (2.3.16)
with a change in notation, |A ij | n = 0, that is, adj A = 0. Hence, the Cauchy
identity is valid for all A.

3.5.3 An Identity Involving a Hybrid Determinant
Let A n = |a ij | n and B n = |b ij | n , and let H ij denote the hybrid determinant
formed by replacing the jth row of A n by the ith row of B n . Then,
n

H ij = b is A js . (3.5.2)
s=1
Theorem.

|a ij x i + b ij | n = A n ij x i + H ij , A n =0.
δ

A n n
In the determinant on the right, the x i appear only in the principal diagonal.
Proof. Applying the Cauchy identity in the form
|A ji | n = A n−1
n
and the formula for the product of two determinants (Section 1.4),
|a ij x i + b ij | n A n−1
n = |a ij x i + b ij | n |A ji | n
= |c ij | n ,
where
n

c ij = (a is x i + b is )A js
s=1
n n

a is A js + b is A js
= x i
s=1 s=1
= δ ij A n x i + H ij .