Putnam 2015 B3

Let $S$ be the set of all $2 \times 2$ real matrices $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$.