Question

线性代数：计算下列n阶行列式(3),

Answer 1

解: D =

ri-r(i+1), i=1,2,...,n-1

0 1 1 ... 1 1

0 1-x 1 ... 1 1

0 0 1-x ... 1 1

... ...

0 0 0 ...1-x 1

1 x x ... x 1

按第1列展开

D = (-1)^(n+1)*

1 1 ... 1 1

1-x 1 ... 1 1

0 1-x ... 1 1

.. ...

0 0 ...1-x 1

ri-r(i+1), i=1,2,...,n-2

x 0 ... 0 0

1-x x ... 0 0

0 1-x ... 0 0

... ...

0 0 ...1-x 1

D = (-1)^(n+1) x^(n-2).