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\begin{document}
\noindent Name: \rule{7cm}{.2pt} \vspace{.3cm} \hfill Date:  \rule{3.5cm}{.2pt}
\vspace{.3cm}

\noindent\hrulefill
\begin{center}
{\bf \Large{MA 162}} \\\vspace{.1cm} Week 7 Recitation Worksheet (Thursday)
\end{center}
\vspace{-.3cm}
\hrulefill

\begin{center}
\fbox{\textbf{You must show all work to receive full credit.}}
\end{center}

\begin{enumerate}[ref=\#\arabic*]
%1
\item Solve the following system of equations using the Gauss-Jordan method. 
$$\left\{ \begin{array}{rcrcrcr}
-3x & + & 4y & - & 7z & = & -5\\
x & - & 2y & + & 3z & = & 6\\
3x &  &  & + & z & = & -7
\end{array} \right. $$


\newpage
%2
\item Solve the following system of equations using the Gauss-Jordan method. 
$$\left\{ \begin{array}{rcrcrcr}
x & - & 6y & - & 4z & = & 5\\
5x & + & 4y & + & z & = & -2\\
7x & - & 8y & - & 7z & = & 1
\end{array} \right. $$


\newpage
%3
\item Solve the following system of equations using the Gauss-Jordan method. 
$$\left\{ \begin{array}{rcrcrcrcr}
w & - & 5x & + & 2y & + & 3z & = & 0\\
-3w & + & 15x & - & 5y & - & 9z & = & -4\\
2w & - & 10x & + & y & + & 6z & = & 12
\end{array} \right. $$


\newpage
%4
\item The Gauss-Jordan method was applied to three different systems of equations to obtain the following augmented matrices in reduced row echelon form.  Determine how many solutions each system has, and justify your answers.
\begin{parts}
\part $\left[ \begin{array}{rrr|r}
  1 & -6 & 0 & 0 \\
  0 & 0 & 1 & 0\\
  0 & 0 & 0 & 1
\end{array} \right]$\vfill
\part $\left[ \begin{array}{rrr|r}
  1 & 0 & -5 & 2 \\
  0 & 1 & 3 & -4\\
  0 & 0 & 0 & 0
\end{array} \right]$\vfill
\part $\left[ \begin{array}{rrr|r}
  1 & 0 & 0 & 2 \\
  0 & 1 & 0 & -4\\
  0 & 0 & 1 & 8\\
  0 & 0 & 0 & 0
\end{array} \right]$\vfill\vspace{1cm}
\end{parts}


%5
\item Let $A = \left[\begin{array}{rr}3 & -5\\k & 8\end{array}\right]$.
\begin{parts}
\item For which value(s) of $k$ is $A$ non-invertible?\vfill\vfill
\item For which value(s) of $k$ is $A$ invertible?\vfill
\end{parts}
\end{enumerate}
\end{document}
