\documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz,kgm}

\newcommand{\kgmcomment}[1]{}

\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

\begin{document}
\headers{Math343}{Quiz 2}{Sep 5, 2025}

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\noindent
\hfill Name: \underline{\hspace{3.5in}} 

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\newcommand{\kgmarray}[2]%
{% 
    \begin{array}{*{#1}{r}}
        #2
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{%
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        \scriptsize
        \node[rectangle,align=left] (TMP) {#1} ;
        \path (TMP.south west) ++(-0.3ex, -0.3ex) node[coordinate] (TMPLEFT) {} ;
        \path (TMP.south east) ++(0.3ex, -0.3ex) node[coordinate] (TMPRIGHT) {} ;
        \draw[->] (TMPLEFT) -- (TMPRIGHT) ;
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\noindent
\textbf{Directions:} Show all work.  No credit for answers without work.

\begin{enumerate}

\item\points{6} Find the general solution of the system with the following augmented matrix.
\[
\kgmmatrix{6}%
{
    3 &  0  &  -3  &  6  & -2  &  -3\\
    3 &  0  &  -3  &  6  & -3  &  -6\\
   10 &  1  &  -6  & 21  & -3  &   3\\
   -8 &  0  &   8  &-16  &  3  &   1
}
\]
\vfill

\newpage
\item\ppointsf{2}{1.5} Let $A$ be a $(8\times 4)$-matrix such that each of the four columns is a pivot column.
\begin{enumerate}
    \item Give the \emph{reduced row-echelon form} of $A$.
    \vfill
    
    \item Describe the \emph{row-echelon form(s)} of $A$.
    \vfill
\end{enumerate}

\item\points{1} Let $A$ be an $(m\times n)$ augmented matrix that represents a linear system with a unique solution.  What can be said about the relationship between $m$ and $n$?
\vfill

\end{enumerate}
\end{document}
