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\begin{document}

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~
\vspace{2cm}

\noindent
\textbf{Directions:}  
\begin{enumerate}
	\item Write your name with one character in each box below.
	\item Show all work.  No credit for answers without work.
	\item You are permitted to use one 8.5 inch by 11 inch sheet of prepared notes.  No other aides are allowed.
\end{enumerate}

\newpage
\headers{Math 343}{Test 3}{Nov. 14, 2025}

~
\vspace{1.5cm}
\begin{enumerate}
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% Page 1  Subspaces 
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\item\points{15} Given the matrix $A$ and the reduced row echelon form of $A$, find bases for $\Col(A)$ and $\Nul(A)$.
\[ A = \mtrix{7}{
	0	& -1	& -2	& -5	& 0	& 1	& -1 \\
	0	& -2	& -4	& -10	& -2	& 11	& 14 \\
	0	& 1	& 2	& 5	& 1	& -5	& -6 \\
	0	& 0	& 0	& 0	& 1	& -5	& -9
}
\ldots \to \ldots 
\mtrix{7}{
	0	& 1	& 2	& 5	& 0	& 0	& 3 \\
	0	& 0	& 0	& 0	& 1	& 0	& 1 \\
	0	& 0	& 0	& 0	& 0	& 1	& 2 \\
	0	& 0	& 0	& 0	& 0	& 0	& 0
}.
\]
\vfill

\item\points{10} Let $V$ be the subspace of $\RR^4$ containing the vectors $\mtrix{1}{x_1 \\ x_2 \\ x_3 \\ x_4}$ such that $x_1+x_2+x_3+x_4 = 0$ and $x_1 - 2x_2 + 3x_3 - 4x_4 = 0$.  Find a basis for $V$.
\vfill

%\item Let $W = \Span\{\vv u_1, \vv u_2\}$ where $\vv u_1 = \mtrix{1}{1 \\ 3 \\ 0 \\ -1}$ and $\vv u_2 = \mtrix{1}{2 \\ -1 \\ 1 \\ 1}$.  Find a $2\times 4$ matrix $B$ such that $\Nul(B) = W$.
%\vfill


\newpage
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\vspace{1.5cm}
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% Page 2 Determinants
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\newcommand{\myspc}{\vspace{5cm}\\~}
\newcommand{\mylargespc}{\vspace{8cm}\\~}

\item\ppoints{4}{5} Compute the determinant of the following matrices.
\begin{multicols}{2}
\begin{enumerate}
	\item $\mtrix{2}{3 & 1 \\ -2 & -5}$
	\myspc
	
	\item $\mtrix{4}{0 & 0 & 0 & a \\ b & 0 & c & d \\ e & f & g & h \\ i & 0 & j & k}$
	\mylargespc

	\columnbreak
	\item $\mtrix{3}{1 & 3 & -1 \\ 2 & 1 & -1 \\ 0 & 5 & -1}$
	\myspc
	

	\item $\mtrix{4}{2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2}$
	\mylargespc
\end{enumerate}
\end{multicols}

\item\points{5} Let $A$ be an $n\times n$ matrix such that $A^2 = I_n$.  Prove that $\det(A)$ equals $1$ or $-1$.
\vfill

\newpage
~
\vspace{1.5cm}
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% Page 3  Eigenvectors
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\item\points{15} Diagonalize the following matrix if possible by finding an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$.  There is no need to compute $P^{-1}$.  If diagonalization is not possible, then explain why.
\[ \mtrix{3}{
	-5	& 8	& 0 \\
	-4	& 7	& 0 \\
	8	& -16	& -1
}
\]
\vfill

\item\points{15} Give a formula for $A^k$, where $A = \mtrix{2}{
	-4	& 3 \\
	-6	& 5
}$.
\vfill

\newpage
~
\vspace{1.5cm}
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% Page 4  Misc
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\item\ppoints{5}{3} True/False.  Justify your answer.
\begin{enumerate}
	\item If $A$ and $B$ are $n\times n$ matrices, then $\det(A+B) = \det(A) + \det(B)$.
	\vspace{1.5cm}

	\item Matrices $A$ and $B$ are similar if and only if $A$ and $B$ have the same eigenvalues with the same multiplicities.
	\vspace{1.5cm}

	\item Let $A$ be a matrix with 7 rows and 10 columns.  If the null space of $A$ has dimension $4$, then the column space of $A$ has dimension $6$.
	\vspace{1.5cm}

	\item If $A$ is a square matrix and $r$ is a scalar, then $A$ is similar to $rA$.
	\vspace{1.5cm}
	
	\item If $A$ and $B$ are similar matrices, then $A^{-1}$ and $B^{-1}$ are also similar.
	\vspace{1.5cm}
\end{enumerate}

\item\points{5} Find a matrix $B$ such that $B^2 = \mtrix{2}{-1 & 5 \\ -10 & 14}$.
\vfill

\end{enumerate}

\newpage
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% Page Scratch
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\end{document}
