Exercise 1.6.22

$A$ has eigenvalues of ${\lambda }_1=1$ and ${\lambda }_2=3$, with corresponding eigenvectors $v_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$ and $v_2=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$. So we have

$A=\mathit{X\Lambda }{X}^{-1}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \frac{1}{2}\hfill & \hfill -\frac{1}{2}\hfill \end{array}\right]$

$A^k=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 3^k\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \frac{1}{2}\hfill & \hfill -\frac{1}{2}\hfill \end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}\hfill 1+3^k\hfill & \hfill 1-3^k\hfill \\ \hfill 1-3^k\hfill & \hfill 1+3^k\hfill \end{array}\right]$