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\begin{document}
\headers{Math343}{Quiz 11}{Nov 7, 2025}

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\noindent
\textbf{Directions:} Show all work.  No credit for answers without work.

\begin{enumerate}

\item\points{3} Find a basis for the eigenspace corresponding to the given eigenvalue.
\begin{align*}
\mtrix{3}{
	1	& 0	& -3 \\
	-6	& -2	& 6 \\
	0	& 0	& -2
}, \lambda = -2
\end{align*}
\vfill

\item\ppoints{2}{2} Find the characteristic equation and eigenvalues with multiplicities of the following matrices.
\begin{enumerate}
\item $\mtrix{2}{
	-6	& 5 \\
	-10	& 9
}$
\vfill

\item $\mtrix{3}{
	-8	& 10	& 0 \\
	-5	& 7	& 0 \\
	10	& -10	& 2
}$
\vfill
\end{enumerate}

\newpage
\item\points{1} Let $A$ be an $n\times n$ matrix.  Prove that if $A$ and $I_n$ are similar, then $A=I_n$.
\vspace{5cm}

\item\ppointsf{4}{0.5} True/False.  In the following, $A$ and $B$ are $n\times n$ matrices.  Justify your answers.
\begin{enumerate}
	\item If $A$ and $B$ have the same set of eigenvectors, then $A$ and $B$ are similar.
	\vfill
	\item If $A$ and $B$ are similar, then $A$ and $B$ have the same set of eigenvectors.
	\vfill
	\item If $A$ and $B$ have the same characteristic polynomial, then $A$ and $B$ are similar.
	\vfill
	\item If $A$ and $B$ are similar, then $A$ and $B$ have the same characteristic polynomial.
	\vfill
\end{enumerate}


\end{enumerate}
\end{document}
