1. 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 $|A|=0$?

2. Substract the first row from the others:

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

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