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\begin{document}

\begin{center}
\textbf{Math 3013} \\[0pt]
FIRST EXAM \\[0pt]
12:30 - 1:20pm, September 15, 2021
\end{center}

\begin{flushright}
Name:\underline{\hspace{2.5 in}}
\end{flushright}

1. Let 
\begin{equation*}
\quad \mathbf{A}=\left[ 
\begin{array}{ccc}
1 & 0 & -1 \\ 
0 & 1 & 0 \\ 
2 & 1 & 1%
\end{array}%
\right] \quad ,\quad \mathbf{B}=\left[ 
\begin{array}{ccc}
2 & 0 & 1 \\ 
0 & -1 & 0%
\end{array}%
\right]
\end{equation*}%
Compute the matrix products $\mathbf{AB}$ and $\mathbf{BA}$ (if they exist)

\vspace{2in}

2. For each of the following augmented matrices, indicate

\begin{itemize}
\item the number of equations and the number of variables in the
corresponding linear system

\item whether or not the corresponding linear system has a solution

\item if the corresponding linear system does have a solution, the number of
free variables in the solution.
\end{itemize}

Hint: note that these augmented matrices are already in row echelon form and
you do not have to actually solve the systems.

(a) $\left[ 
\begin{array}{ccc|c}
1 & 1 & 0 & 3 \\ 
0 & 1 & 2 & 2 \\ 
0 & 0 & 1 & 1 \\ 
0 & 0 & 0 & 1%
\end{array}%
\right] $

\vspace{.7in}

(b) $\left[ 
\begin{array}{cccc|c}
1 & 0 & 0 & 2 & 1 \\ 
0 & 0 & 1 & 1 & 2 \\ 
0 & 0 & 0 & 0 & 0%
\end{array}%
\right] $

\vspace{.7in}

(c) $\left[ 
\begin{array}{ccc|c}
1 & -1 & 2 & 2 \\ 
0 & 2 & 1 & 1 \\ 
0 & 0 & 3 & 1%
\end{array}%
\right] $

\newpage

3. Consider the following linear system 
\begin{equation*}
\begin{array}{rcr}
x_{2}+2x_{3}+x_{4} & = & 1 \\ 
2x_{1}+x_{2}+x_{3}+2x_{4} & = & -1 \\ 
x_{1}+x_{2}+2x_{3}-x_{4} & = & 3%
\end{array}%
\end{equation*}%
Write down the corresponding augmented matrix and row reduce it to
row-echelon form.

\vspace{3.5in}

4. Row reduce the following augmented matrix to \textbf{reduced} row-echelon
form. 
\begin{equation*}
\left[ \mathbf{A}\mid \mathbf{b}\right] =\left[ 
\begin{array}{ccccc|c}
1 & 2 & 1 & 2 & 2 & 2 \\ 
0 & 0 & 2 & 4 & 4 & 4 \\ 
0 & 0 & 0 & 1 & -2 & 2%
\end{array}%
\right] 
\end{equation*}

\newpage

5. Suppose the augmented matrix below is the Reduced Row Echelon Form of an
augmented matrix of a linear system. Display the solution of the linear
system as a hyperplane (within the space of variables). 
\begin{equation*}
\left[ 
\begin{array}{ccccc|c}
1 & 0 & 0 & -2 & 0 & 1 \\ 
0 & 0 & 1 & 1 & 0 & 2 \\ 
0 & 0 & 0 & 0 & 1 & 1%
\end{array}%
\right] 
\end{equation*}

\vspace{3.5in}

6. Determine if the matrix below has an inverse and, if so, determine its
inverse. (Hint: a single row reduction process will answer both parts of to
this problem.) 
\begin{equation*}
\mathbf{A}=\left[ 
\begin{array}{ccc}
1 & -1 & 1 \\ 
-2 & 2 & -1 \\ 
1 & 0 & 0%
\end{array}%
\right] 
\end{equation*}

\end{document}
