Determinants and Their Applications in Mathematical Physics

4.1 Alternants 55

Hence,

[xij]n[Fji]n=I,

[Fji]n=[xij]

− 1

=[X

ji

n

]n.

The theorem follows.

Theorem 4.2.

X

(n)

nj =(−1)

n−j

Xn− 1 σ

(n−1)

n−j.

Proof. Referring to equations (A.7.1) and (A.7.3) in Appendix A.7,

Xn=Xn− 1

n− 1

∏

r=1

(xn−xr)

=Xn− 1 fn− 1 (xn)

=Xn− 1 gnn(xn).

From Theorem 4.1,

X

(n)

nj

=

(−1)

n−j

Xnσ

(n)

n,n−j

gnn(xn)

=(−1)

n−j

Xn− 1 σ

(n)

n,n−j.

The proof is completed using equation (A.7.4) in Appendix A.7.

4.1.4 A Hybrid Determinant

LetYnbe a second Vandermondian defined as

Yn=|y

j− 1

i

|n

and letHrsdenote the hybrid determinant formed by replacing therth

row ofXnby thesth row ofYn.

Theorem 4.3.

Hrs

Xn

=

gnr(ys)

gnr(xr)

.

Proof.

Hrs

Xn

=

n

∑

j=1

y

j− 1

s X

rj

n

=

1

gnr(xr)

n

∑

j=1

(−1)

n−j

σ

(n)

r,n−j

y

j− 1

s

(Putj=n−k)

=

1

gnr(xr)

n− 1

∑

k=0

(−1)

k

σ

(n)

rk

y

n− 1 −k

s

.