I'm trying to show:
A "magic square" $A$ is a matrix $n\times n$ with slots $1,2,\cdots, n^2$ such that the sum of the elements of each row (and column) is the same . Prove that $\frac{n(n^2+1)}{2}$ is a eigenvalue of the matrix $A$.
I was trying to make a proof with a proposition: "$\beta$ is a eigenvalue of $A$ if and only if $\det(A-\beta I_n)=0$", I is the matrix idetity $n\times n$. But I can not do it.
Thanks for your help.