Given A whose factorization $PD{P}^{-1}$ is shown below, write an expresssion for $A^k$ where $k$ represents an

abitrary positive integer. The answer should written as a single matrix.

$A=\left[\begin{array}{cc}7& -20\\ 1& -2\end{array}\right]=\left[\begin{array}{cc}5& 4\\ 1& 1\end{array}\right]\left[\begin{array}{cc}3& 0\\ 0& 2\end{array}\right]\left[\begin{array}{cc}1& -4\\ -1& 5\end{array}\right].$

$(5$ pts$)$ Given A whose factorization $PD{P}^{-1}$ is shown below, find the eigenvalues of A and a basis for each

eigenspace.

$A=\left[\begin{array}{ccc}-16& 9& 9\\ -6& 5& 3\\ -36& 18& 20\end{array}\right]=\left[\begin{array}{ccc}2& 1& 3\\ 1& 1& 1\\ 3& 1& 6\end{array}\right]\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 5\end{array}\right]\left[\begin{array}{ccc}5& -3& -2\\ -3& 3& 1\\ -2& 1& 1\end{array}\right]$