Determinants and Their Applications in Mathematical Physics

66 4. Particular Determinants

=|cij|n,

where

cij=

n

∑

r=1

(−1)

r+1

an+1−i,ran+1−j,n+1−r (putr=n+1−s)

=(−1)

n+1

n

∑

s=1

(−1)

s+1

an+1−j,san+1−i,n+1−s

=(−1)

n+1

cji. (4.3.4)

The theorem follows.

Theorem 4.9. A skew-symmetric determinant of odd order is identically

zero.

Proof. LetA

∗

2 n− 1 denote the determinant obtained from A^2 n−^1 by

changing the sign of every element. Then, since the number of rows and

columns is odd,

A

∗

2 n− 1

=−A 2 n− 1.

But,

A

∗

2 n− 1 =A

T

2 n− 1 =A^2 n−^1.

Hence,

A 2 n− 1 =0,

which proves the theorem.

The cofactorA

(2n)

ii is also skew-symmetric of odd order. Hence,

A

(2n)

ii

=0. (4.3.5)

By similar arguments,

A

(2n)

ji

=−A

(2n)

ij

,

A

(2n−1)

ji

=A

(2n−1)

ij

. (4.3.6)

It may be verified by elementary methods that

A 2 =a

2

12 , (4.3.7)

A 4 =(a 12 a 34 −a 13 a 24 +a 14 a 23 )

2

. (4.3.8)

Theorem 4.10. A 2 n is the square of a polynomial function of its

elements.

Proof. Use the method of induction. Applying the Jacobi identity

(Section 3.6.1) to the zero determinantA 2 n− 1 ,

∣

∣

∣

∣

∣

A

(2n−1)

ii

A

(2n−1)

ij

A

(2n−1)

ji

A

(2n−1)

jj

∣

∣

∣

∣

∣

=0,