Page 148 - Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 133

= c ij .
2i − 1
After removing the factor (2i − 1) −1 from row i,1 ≤ i ≤ n, the result is
x

x
3

x

n 5
U = 2 n! [c ij ] n .
(2n)! ···
x
2n−1

111 ··· 1 •

n+1
Transposing,

1

1

n
U = 2 n! [−c ij ] n 1 .
(2n)! ···

1
xx x ··· x •
3 5 2n−1
n+1
Now, change the signs of columns 1 to n and row (n + 1). This introduces
(n + 1) negative signs and gives the result
2 n!
(−1) n+1 n
U = Z. (4.10.27)
(2n)!
Perform the column operations
C = C j + C n+1 , 1 ≤ j ≤ n,

j
on V . The result is that [a ij ] n is replaced by [a ] n , where
∗
ij
1
a = a ij + .
∗
2i − 1
ij
Perform the row operations
x 2i−1

R = R i − R n+1 , 1 ≤ i ≤ n,
2i − 1
i
∗∗
which results in [a ] n being replaced by [a ] n , where
∗
ij ij
x 2(i+j−1)
∗
a ∗∗ = a −
2i − 1
ij ij
= c ij .
2i − 1
After removing the factor (2i − 1) −1 from row i,1 ≤ i ≤ n, the result is
2 n!
n
V = Z. (4.10.28)
(2n)!
The theorem follows from (4.10.27) and (4.10.28).