Directions: You may work to solve these problems in groups, but all written work must be your own. Show all work; no credit for solutions without work..

1. [5.3.4] Given the factorization $A=\mathit{𝑃𝐷}{P}^{-1}$ below, find a formula for $A^k$ where $k$ is a nonnegative integer.

   $$\left[\begin{array}{cc}\hfill -6& \hfill 8\\ \hfill -4& \hfill 6\end{array}\right]=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 1& \hfill 1\end{array}\right]\left[\begin{array}{cc}\hfill 2& \hfill 0\\ \hfill 0& \hfill -2\end{array}\right]\left[\begin{array}{cc}\hfill -1& \hfill 2\\ \hfill 1& \hfill -1\end{array}\right]$$

2. [5.3.{7-20}] Diagonalize the following matrices if possible. That is, for each diagonalizable matrix $A$ below, construct an invertible matrix $P$ and a diagonal matrix $D$ such that $A=\mathit{𝑃𝐷}{P}^{-1}$. (There is no need to compute $P^{-1}$ explicitly.) For each matrix $A$ below that is not diagonalizable, explain why not.

   1. $\left[\begin{array}{cc}\hfill 1& \hfill 0\\ \hfill 6& \hfill -1\end{array}\right]$
   2. $\left[\begin{array}{cc}\hfill 3& \hfill -1\\ \hfill 1& \hfill 5\end{array}\right]$
   3. $\left[\begin{array}{ccc}\hfill -1& \hfill 4& \hfill -2\\ \hfill -3& \hfill 4& \hfill 0\\ \hfill -3& \hfill 1& \hfill 3\end{array}\right]$
   4. $\left[\begin{array}{ccc}\hfill 4& \hfill 0& \hfill 2\\ \hfill 2& \hfill 3& \hfill 4\\ \hfill 0& \hfill 0& \hfill 3\end{array}\right]$
   5. $\left[\begin{array}{cccc}\hfill 2& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 2& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 2& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0& \hfill 2\end{array}\right]$

3. [5.3.{21-28}] True/False. In the following, $A$, $P$, and $D$ are $(n\times n)$ matrices. Justify your answers.

   1. $A$ is diagonalizable if $A=\mathit{𝑃𝐷}{P}^{-1}$ for some matrix $D$ and some invertible matrix $P$.
   2. If ${ℝ}^n$ has a basis of eigenvectors of $A$, then $A$ is diagonalizable.
   3. $A$ is diagonalizable if and only if $A$ has $n$ eigenvalues, counting multiplicities.
   4. If $A$ is diagonalizable, then $A$ is invertible.
   5. $A$ is diagonalizable if $A$ has $n$ eigenvectors.
   6. If $A$ is diagonalizable, then $A$ has $n$ distinct eigenvalues.
   7. If $\mathit{𝐴𝑃}=\mathit{𝑃𝐷}$, with $D$ diagonal, then the nonzero columns of $P$ must be eigenvectors of $A$.
   8. If $A$ is invertible, then $A$ is diagonalizable.
4. [5.3.31] A is a $4\times 4$ matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two-dimensional. Is it possible that $A$ is not diagonalizable? Justify your answer.
5. Let ${𝐯}_1=\left[\begin{array}{c}\hfill 3\\ \hfill 1\end{array}\right]$ and ${𝐯}_2=\left[\begin{array}{c}\hfill 2\\ \hfill 7\end{array}\right]$, and let ${\lambda }_1$ and ${\lambda }_2$ be scalars. Construct a $(2\times 2)$-matrix $A$ having eigenvectors ${𝐯}_1$ and ${𝐯}_2$ with respective eigenvalues ${\lambda }_1$ and ${\lambda }_2$. (Here, the entries of $A$ will depend on ${\lambda }_1$ and ${\lambda }_2$.)