Let $$ S = \left\{ \begin{pmatrix} x & y \\ z & w \end{pmatrix} \in \mathbb{R}^{2\times2}: \; x,y,z,w \; \text{ is an arithmetic sequence}\right\}. $$ Find all matrices $C \in S$ that satisfying $\;\exists k \in \mathbb{Z}, k\geq 2, C^k \in S$.
It is clear that $$ x\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} $$ satisfies the problem, and it seems that it is the only solution. We then assume that the common difference is not zero. We can easily find that we only need to consider $$ \begin{pmatrix} -3 & -1 \\ 1 & 3 \end{pmatrix} + r\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}. $$ But I don't know what should I do next. By the way I find that $k$ is odd but I can't make use of it.