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),