66 2 Algebra
206.Prove that
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(x^2 + 1 )^2 (xy+ 1 )^2 (xz+ 1 )^2
(xy+ 1 )^2 (y^2 + 1 )^2 (yz+ 1 )^2
(xz+ 1 )^2 (yz+ 1 )^2 (z^2 + 1 )^2∣
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= 2 (y−z)^2 (z−x)^2 (x−y)^2.207.Let(Fn)nbe the Fibonacci sequence. Using determinants, prove the identity
Fn+ 1 Fn− 1 −Fn^2 =(− 1 )n, for alln≥ 1.208.Letp<mbe two positive integers. Prove that
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(m
0)(m
1)
···
(m
p)
(m+ 1
0)(m+ 1
1)
···
(m+ 1
p)
..
.
..
.
... ..
(.
m+p
0)(m+p
1)
···
(m+p
p)
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= 1.
209.Given distinct integersx 1 ,x 2 ,...,xn, prove that
∏
i>j(xi−xj)is divisible by
1! 2 !···(n− 1 )!.210.Prove the formula for the determinant of a circulant matrix
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x 1 x 2 x 3 ···xn
xnx 1 x 2 ···xn− 1
..
...
.
..
.
.....
.
x 3 x 4 x 5 ···x 2
x 2 x 3 x 4 ···x 1∣
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=(− 1 )n−^1n∏− 1j= 0( n
∑k= 1ζjkxk)
,
whereζ=e^2 πi/n.- Compute the determinant of then×nmatrixA=(aij)ij, where
aij={
(− 1 )|i−j| ifi =j,
2ifi=j.212.Prove that for any integersx 1 ,x 2 ,...,xnand positive integersk 1 ,k 2 ,...,kn, the
determinant
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x 1 k^1 x 2 k^1 ···xnk^1
x 1 k^2 x 2 k^2 ···xnk^2
..
...
.
... ..
.
x 1 knx 2 kn···xnkn∣
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is divisible byn!.