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 (adjA)
∗

. Then,

(adjA)

∗

=σadjA,

where

σ=(−1)

(p 1 −1)+(p 2 −2)+···+(pr−r)+(q 1 −1)+(q 2 −2)+···+(qr−r)

=(−1)

(p 1 +p 2 +···+pr)+(q 1 +q 2 +···+qr)

.

Now replace eachAijin (adjA)

∗

byaij, transpose, and denote the result

by|aij|

∗

. Then,

|aij|

∗

=σ|aij|=σA.

Raise the order ofJfromr tonin a manner similar to that shown in

(3.6.3), augmenting the firstrcolumns until they are identical with the

firstrcolumns of (adjA)

∗

, denote the result byJ

∗

, and form the product

|aij|

∗

J

∗

. The theorem then appears.

Illustration.Let (n, r)=(4,2) and let

J=J 23 , 24 =

∣

∣

∣

∣

A 22 A 32

A 24 A 34

∣

∣

∣

∣

.

Then

(adjA)

∗

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

A 22 A 32 A 12 A 42

A 24 A 34 A 14 A 44

A 21 A 31 A 11 A 41

A 23 A 33 A 13 A 43

∣ ∣ ∣ ∣ ∣ ∣ ∣

=σadjA,

where

σ=(−1)

2+3+2+4

=− 1

and

|aij|

∗

=

∣

∣

∣

∣

∣

∣

∣

a 22 a 24 a 21 a 23

a 32 a 34 a 31 a 33

a 12 a 14 a 11 a 13

a 42 a 44 a 41 a 43

∣

∣

∣

∣

∣

∣

∣

=σ|aij|=σA.

The first two columns ofJ

∗

are identical with the first two columns of

(adjA)

∗

:

J=J

∗

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

A 22 A 32

A 24 A 34

A 21 A 31 1

A 23 A 33 1

∣ ∣ ∣ ∣ ∣ ∣ ∣

,

σAJ=|aij|

∗

J

∗