Page 81 - Determinants and Their Applications in Mathematical Physics

66 4. Particular Determinants

= |c ij | n ,
where
n
r+1
c ij = (−1) a n+1−i,r a n+1−j,n+1−r (put r = n +1 − s)
r=1
n
s+1
n+1
=(−1) (−1) a n+1−j,s a n+1−i,n+1−s
s=1
=(−1) n+1 c ji . (4.3.4)
The theorem follows.

Theorem 4.9. A skew-symmetric determinant of odd order is identically
zero.
Proof. Let A ∗ denote the determinant obtained from A 2n−1 by
2n−1
changing the sign of every element. Then, since the number of rows and
columns is odd,
A ∗ = −A 2n−1 .
2n−1
But,

A ∗ = A T = A 2n−1 .
2n−1 2n−1
Hence,
A 2n−1 =0,
which proves the theorem.
(2n)
The cofactor A is also skew-symmetric of odd order. Hence,
ii
(2n)
A =0. (4.3.5)
ii
By similar arguments,
(2n) (2n)
A = −A ,
ji ij
(2n−1) (2n−1)
A = A . (4.3.6)
ji ij
It may be verified by elementary methods that
2
A 2 = a , (4.3.7)
12
2
A 4 =(a 12 a 34 − a 13 a 24 + a 14 a 23 ) . (4.3.8)
Theorem 4.10. A 2n 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 determinant A 2n−1 ,
(2n−1)
A A (2n−1)

ii ij =0,
(2n−1) (2n−1)
A A
ji jj