Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 15

2.3.6 The Cofactors of a Zero Determinant........

IfA= 0, then

Ap 1 q 1 Ap 2 q 2 =Ap 2 q 1 Ap 1 q 2 , (2.3.16)

that is,

∣

∣

∣

∣

Ap

1 q 1

Ap

1 q 2

Ap

2 q 1

Ap

2 q 2

∣

∣

∣

∣

=0, 1 ≤p 1 ,p 2 ,q 1 ,q 2 ≤n.

It follows that
∣
∣
∣
∣
∣
∣

Ap

1 q 1

Ap

1 q 2

Ap

1 q 3

Ap

2 q 1

Ap

2 q 2

Ap

2 q 3

Ap 3 q 1 Ap 3 q 2 Ap 3 q 2

∣

∣

∣

∣

∣

∣

=0

since the second-order cofactors of the elements in the last (or any) row are

all zero. Continuing in this way,

∣ ∣ ∣ ∣ ∣ ∣ ∣

Ap

1 q 1

Ap

1 q 2

··· Ap

1 qr

Ap

2 q 1

Ap

2 q 2

··· Ap

2 qr

··· ··· ··· ···

Ap

rq 1

Ap

rq 2

··· Ap

rqr

∣ ∣ ∣ ∣ ∣ ∣ ∣ r

=0, 2 ≤r≤n. (2.3.17)

This identity is applied in Section 3.6.1 on the Jacobi identity.

2.3.7 The Derivative of a Determinant

If the elements ofAare functions ofx, then the derivative ofAwith respect

toxis equal to the sum of thendeterminants obtained by differentiating

the columns ofAone at a time:

A

′

=

n

∑

j=1

∣

∣C

1 C 2 ···C

′

j

···Cn

∣

∣

=

n

∑

i=1

n

∑

j=1

a

′

ijAij. (2.3.18)