Chapter 3.11, Problem 59P

Repeat the last part of Problem 58 for the matrix $M=\left(\begin{array}{cc}3& -1\\ -1& 3\end{array}\right).$

Note in Section 6 [see (6.15)) that, for the given matrix A, we found ${\text{A}}^2=-\text{I,}$ so it was easy to find all the powers of A. It is not usually this easy to find high powers of a matrix directly. Try it for the square matrix M in equation $\left(11.1\right).$ Then use the method outlined in Problem 57 to find ${\text{M}}^4,{\text{M}}^{10},e^{\text{M}}.$