Let us define a magic square as a matrix whose entries are rational numbers and have the same sum on columns, rows and both main diagonals. I am trying to prove the exercise :
For every $n\ge4$, there is a $n×n$ magic square $A$ such that for every integer $m\ge2$, $A^m$ is not a magic square.
I have found examples for even $n\ge4$ and odd $n\ge7$, respectively of the form: $$ \begin{pmatrix} n-2 & 0\\ 0 & S-E \end{pmatrix} $$
$$ \begin{pmatrix} n-3 & 0\\ 0 & S-E-D \end{pmatrix} $$
where $S$ is the matrix with all entries equal to $1$, $E$ is the identity matrix, $D$ is the matrix whose second diagonal are the $1$s and all other $0$.
What's left is the $n=5$ case. Please show me an example of that or even better, a single one covers all the cases.Thank you in advance.