Page 55 - Determinants and Their Applications in Mathematical Physics

40 3. Intermediate Determinant Theory

as a block in the top left-hand corner. Denote the result by (adj A) . Then,
∗
∗
(adj A) = σ adj A,
where
σ =(−1) (p 1 −1)+(p 2 −2)+···+(p r −r)+(q 1 −1)+(q 2 −2)+···+(q r −r)
=(−1) (p 1 +p 2 +···+p r )+(q 1 +q 2 +···+q r ) .
∗
Now replace each A ij in (adj A) by a ij , transpose, and denote the result
∗
by |a ij | . Then,
∗
|a ij | = σ|a ij | = σA.
Raise the order of J from r to n in a manner similar to that shown in
(3.6.3), augmenting the first r columns until they are identical with the
first r columns of (adj A) , denote the result by J , and form the product
∗
∗
|a ij | J . The theorem then appears.
∗
∗
Illustration. Let (n, r)=(4, 2) and let

A 22
J = J 23,24 = A 32 .
A 24 A 34

Then

A 22 A 32 A 12 A 42

∗ A 24 A 34 A 14 A 44
(adj A) =
A 21 A 31 A 11 A 41

A 23 A 33 A 13 A 43
= σ adj A,
where
2+3+2+4
σ =(−1) = −1
and

a 22 a 24 a 21 a 23

∗ a 32 a 34 a 31 a 33

|a ij | =
a 12 a 14 a 11 a 13

a 42 a 44 a 41 a 43
= σ|a ij | = σA.
The first two columns of J ∗ are identical with the first two columns of
∗
(adj A) :

A 22 A 32

∗ A 24 A 34 ,

1
J = J =
A 21 A 31
1
A 23 A 33
σAJ = |a ij | J ∗
∗