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\headers{Math343}{Homework 11}{Due Nov 5, 2025}

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\noindent
\textbf{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..

\begin{enumerate}

\item {} [5.1.\{9-16\}] Find a basis for the eigenspace corresponding to each listed eigenvalue.
\begin{multicols}{2}
\begin{enumerate}
	\item $\mtrix{2}{5 & 0 \\ 2 & 1}$, $\lambda=1,5$
	\item $\mtrix{2}{4 & -2 \\ -3 & 9}$, $\lambda=10$
	\item $\mtrix{3}{4 & 0 & 1\\ -2 & 1 & 0 \\ -2 & 0 & 1}$, $\lambda=1,2,3$
	\item $\mtrix{3}{4 & 2 & 3\\ -1 & 1 & -3\\ 2 & 4 & 9}$, $\lambda=3$
\end{enumerate}
\end{multicols}

\item True/False.  Justify your answers.  In the following, $A$ is an $(n\times n)$-matrix.
\begin{enumerate}
	\item If $A\vv x=\lambda \vv x$ for some vector $\vv x$, then $\lambda$ is an eigenvalue of $A$.
	\item The matrix $A$ is invertible if and only if $0$ is an eigenvalue of $A$.
	\item To find the eigenvalues of $A$, reduce $A$ to echelon form.
	\item If $\vv v$ is an eigenvector with eigenvalue $2$, then $2\vv v$ is an eigenvector with eigenvalue $4$.
	\item An eigenspace of $A$ is a null space of a certain matrix.
\end{enumerate}

\item {} [5.1.33] Let $\lambda$ be an eigenvalue of an invertible matrix $A$.  Show that $\lambda^{-1}$ is an eigenvalue of $A^{-1}$.

% 5.2 problems

\item {} [5.2.\{1-14\}] Find the characteristic polynomial and eigenvalues of the matrices below.  
\begin{multicols}{2}
\begin{enumerate}
	\item $\mtrix{2}{2 & 7\\ 7 & 2}$
	\item $\mtrix{2}{4 & -3\\ -4 & 2}$
	\item $\mtrix{3}{1 & 0 & -1 \\ 2 & 3 & -1 \\ 0 & 6 & 0}$
	\item $\mtrix{3}{6 & -2 & 0 \\ -2 & 9 & 0 \\ 5 & 8 & 3}$
\end{enumerate}
\end{multicols}

\item {} [5.2.18] Find $h$ in the matrix $A$ below such that the eigenspace for $\lambda=5$ is two-dimensional.
\[ \mtrix{4}{5 & -2 & 6 & -1 \\ 0 & 3 & h & 0 \\ 0 & 0 & 5 & 4 \\ 0 & 0 & 0 & 1}\]

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



\end{enumerate}
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