Name:      

Directions: Show all work. No credit for answers without work.

1. [3 points] Find a basis for the eigenspace corresponding to the given eigenvalue.

   $$\begin{array}{lll}\hfill \left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill -3\\ \hfill -6& \hfill -2& \hfill 6\\ \hfill 0& \hfill 0& \hfill -2\end{array}\right],\lambda =-2& & \hfill \end{array}$$

2. [2 parts, 2 points each] Find the characteristic equation and eigenvalues with multiplicities of the following matrices.

   1. $\left[\begin{array}{cc}\hfill -6& \hfill 5\\ \hfill -10& \hfill 9\end{array}\right]$
   2. $\left[\begin{array}{ccc}\hfill -8& \hfill 10& \hfill 0\\ \hfill -5& \hfill 7& \hfill 0\\ \hfill 10& \hfill -10& \hfill 2\end{array}\right]$

3. [1 point] Let $A$ be an $n\times n$ matrix. Prove that if $A$ and $I_n$ are similar, then $A=I_n$.

4. [4 parts, 0.5 points each] True/False. In the following, $A$ and $B$ are $n\times n$ matrices. Justify your answers.

   1. If $A$ and $B$ have the same set of eigenvectors, then $A$ and $B$ are similar.
   2. If $A$ and $B$ are similar, then $A$ and $B$ have the same set of eigenvectors.
   3. If $A$ and $B$ have the same characteristic polynomial, then $A$ and $B$ are similar.
   4. If $A$ and $B$ are similar, then $A$ and $B$ have the same characteristic polynomial.