Determinant 2

**james_bond** · Mar 28th 2009, 12:50 AM

$|A|=\begin{vmatrix}1&1&1&\dots &1\\1& 1-x& 1&\dots &1\\1&1& 2-x &\dots &1\\\vdots&&&&\\1&1&1&\dots &n-x\end{vmatrix}$

For which $x$ does $|A|=0$ ?

**red_dog** · Mar 28th 2009, 01:34 AM

Substract the first row from the others:

$|A|=\begin{vmatrix}1 & 1 & 1 & \dots & 1\\ 0 & -x & 0 & \ldots & 0\\ 0 & 0 & 1-x & \ldots & 0\\ \vdots\\ 0 & 0 & 0 & \ldots & n-1-x\end{vmatrix}= -x(1-x)(2-x)\ldots (n-1-x)$

$|A|=0\Rightarrow x_1=0, \ x_2=1, \ldots,x_n=n-1$

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