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Math Help - Determinant 2

  1. #1
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    Determinant 2

    |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?
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  2. #2
    MHF Contributor red_dog's Avatar
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    Substract the first row from the others:

    |A|=\begin{vmatrix}1 & 1 & 1 & \dots & 1\\<br />
0 & -x & 0 & \ldots & 0\\<br />
0 & 0 & 1-x & \ldots & 0\\<br />
\vdots\\<br />
0 & 0 & 0 & \ldots & n-1-x\end{vmatrix}=<br />
-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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