Page 70 - Determinants and Their Applications in Mathematical Physics

4.1 Alternants 55

Hence,
[x ij ] n [F ji ] n = I,
[F ji ] n =[x ij ] −1
=[X ] n .
ji
n
The theorem follows.
Theorem 4.2.
(n) (n−1)
X =(−1) n−j X n−1 σ .
nj n−j
Proof. Referring to equations (A.7.1) and (A.7.3) in Appendix A.7,
n−1

(x n − x r )
X n = X n−1
r=1
= X n−1 f n−1 (x n )
= X n−1 g nn (x n ).
From Theorem 4.1,
(n)
(−1) n−j X n σ
(n)
X = n,n−j
g nn (x n )
nj
(n)
=(−1) n−j X n−1 σ .
n,n−j
The proof is completed using equation (A.7.4) in Appendix A.7.
4.1.4 A Hybrid Determinant
Let Y n be a second Vandermondian defined as
j−1
Y n = |y
i | n
and let H rs denote the hybrid determinant formed by replacing the rth
row of X n by the sth row of Y n .
Theorem 4.3.
g nr (y s )
= .
H rs
g nr (x r )
X n
Proof.
n
j−1
= y X rj
H rs
s n
j=1
X n
1 (n)
n
= (−1) n−j σ y j−1 (Put j = n − k)
g nr (x r ) r,n−j s
j=1
n−1
1
= (−1) σ y .
k (n) n−1−k
g nr (x r ) rk s
k=0