<s>[INST] What is the value of the following determinant? \begin{vmatrix}
1 & n & n & \dots & n \\ 
n & 2 & n & \dots & n \\
n & n & 3 & \dots & n \\
\vdots & \vdots& \vdots & \ddots & \vdots\\
n & n & n &   & n\\
\end{vmatrix}
I've the feeling that I should to eliminate with the last row the others ove and after that should I multiply with the elements of the main diagonal.
 [/INST] By Gaussian elimination,
$$\begin{vmatrix}
1 & n & n & \dots & n \\ 
n & 2 & n & \dots & n \\
n & n & 3 & \dots & n \\
\vdots & \vdots& \vdots & \ddots & \vdots\\
n & n & n & \dots  & n\\
\end{vmatrix}=\begin{vmatrix}
1 & n & n & \dots & n \\ 
n-1 & 2-n & 0 & \dots & 0 \\
n-1 & 0 & 3-n & \dots & 0 \\
\vdots & \vdots& \vdots & \ddots & \vdots\\
n-1 & 0 & 0 & \dots  & 0\\
\end{vmatrix}$$
so if we expand along the last column, then along the last row, we get that the determinant equals $\color{red}{n!(-1)^{n+1}}$.
</s>