Determinants and Their Applications in Mathematical Physics

58 4. Particular Determinants

Exercises

1.Prove the reduction formula

A

(n)

ij

=A

(n−1)

ij

n− 1

∏

r=1

r=i

(

xn−xr

xr−yn

)n− 1

∏

s=1

s=j

(

ys−yn

xn−ys

)

.

Hence, or otherwise, prove that

A

ij

n=

1

xi−yj

f(yj)g(xi)

f

′

(xi)g

′

(yj)

,

where

f(t)=

n

∏

r=1

(t−xr),

g(t)=

n

∏

s=1

(t−ys).

2.Let

Vn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

f(x 1 )

f(x 2 )

[aij]n

.

.

.

.

.

.

f(xn)

11 ... ... 11

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

,

Wn=

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣ ∣

f(x 1 )

f(x 2 )

[aij]n

.

.

.

.

.

.

f(xn)

− 1 − 1 ... ... − 11

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

,

where

aij=

(

1 −xiyj

xi−yj

)

f(xi),

f(x)=

n

∏

i=1

(x−yi).

Show that

Vn=(−1)

n(n+1)/ 2

XnYn

n

∏

i=1

(xi−1)(yi+1),