How to calculate this determinant?
$$A=\begin{bmatrix}n-1&k&k&k&\ldots& k\\k&n-1&k&k&\ldots &k\\\ldots&\ldots&\ldots &&\ldots\\\\k&k&k&k&\ldots &n-1\\ \end{bmatrix}_{n\times n}$$
where $n,k\in \Bbb N$ are fixed.
I tried for $n=3$ and got the characteristic polynomial as $(x-2-k)^2(x-2+2k).$
How to find it for general $n\in \Bbb N$?