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. [5.1.{9-16}] Find a basis for the eigenspace corresponding to each listed eigenvalue.

   1. $\left[\begin{array}{cc}\hfill 5& \hfill 0\\ \hfill 2& \hfill 1\end{array}\right]$, $\lambda =1,5$
   2. $\left[\begin{array}{cc}\hfill 4& \hfill -2\\ \hfill -3& \hfill 9\end{array}\right]$, $\lambda =10$
   3. $\left[\begin{array}{ccc}\hfill 4& \hfill 0& \hfill 1\\ \hfill -2& \hfill 1& \hfill 0\\ \hfill -2& \hfill 0& \hfill 1\end{array}\right]$, $\lambda =1,2,3$
   4. $\left[\begin{array}{ccc}\hfill 4& \hfill 2& \hfill 3\\ \hfill -1& \hfill 1& \hfill -3\\ \hfill 2& \hfill 4& \hfill 9\end{array}\right]$, $\lambda =3$

2. True/False. Justify your answers. In the following, $A$ is an $(n\times n)$-matrix.

   1. If $A𝐱=\lambda 𝐱$ for some vector $𝐱$, then $\lambda$ is an eigenvalue of $A$.
   2. The matrix $A$ is invertible if and only if $0$ is an eigenvalue of $A$.
   3. To find the eigenvalues of $A$, reduce $A$ to echelon form.
   4. If $𝐯$ is an eigenvector with eigenvalue $2$, then $2𝐯$ is an eigenvector with eigenvalue $4$.
   5. An eigenspace of $A$ is a null space of a certain matrix.
3. [5.1.33] Let $\lambda$ be an eigenvalue of an invertible matrix $A$. Show that ${\lambda }^{-1}$ is an eigenvalue of $A^{-1}$.

4. [5.2.{1-14}] Find the characteristic polynomial and eigenvalues of the matrices below.

   1. $\left[\begin{array}{cc}\hfill 2& \hfill 7\\ \hfill 7& \hfill 2\end{array}\right]$
   2. $\left[\begin{array}{cc}\hfill 4& \hfill -3\\ \hfill -4& \hfill 2\end{array}\right]$
   3. $\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill -1\\ \hfill 2& \hfill 3& \hfill -1\\ \hfill 0& \hfill 6& \hfill 0\end{array}\right]$
   4. $\left[\begin{array}{ccc}\hfill 6& \hfill -2& \hfill 0\\ \hfill -2& \hfill 9& \hfill 0\\ \hfill 5& \hfill 8& \hfill 3\end{array}\right]$

5. [5.2.18] Find $h$ in the matrix $A$ below such that the eigenspace for $\lambda =5$ is two-dimensional.

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

6. [5.2.20] Use a property of determinants to show that $A$ and $A^T$ have the same characteristic polynomial.