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\begin{document}
\title{314 '09-QUIZ 1}
\author{Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }
\maketitle
\section{$< 10pts >$}
Use Gauss-Jordan elimination to solve the system $A{\bf x} = {\bf b}$ where 
\[ {\bf A} = \left(
\begin{array}{rrrr}
1 & 3 & 1 & 1 \\
2 & -2 & 1 & 2\\
3 & 1 & 2 & -1
\end{array}\right)   \ ,\ 
{\bf b } = \left(
\begin{array}{r}
3 \\ 8\\ -1
\end{array}\right)  \ .
\]
Determine if the system is consistent or inconsistent. If it is
consistent, find all solutions.\\
{\bf Solution}\\
\ \    

\newpage
\section{$< 10pts >$}
Use Gauss-Jordan elimination to find the inverse ${\bf A}^{-1}$ if 
\[ {\bf A} = \left(
\begin{array}{rrr}
1 & 0 & 1 \\2 & 2 & 3\\3 & 3 & 4
\end{array}\right)   \]

{\bf Solution}\\
\ \ 



\newpage
\section{$< 10pts >$}
Consider a linear system whose augmented matrix is of the form
\[ {\bf A} = \left(
\begin{array}{rrr}
1 & 1 & 3 \\1 & 2 & 4\\1 & 3 & a
\end{array}\right|   
\left.
\begin{array}{r}
2 \\ 3 \\ b \end{array} \right) \]
\begin{enumerate}
\item For what values of $a$, $b$ will the system have infinitely many 
solutions?
\item For what values of $a$, $b$ will the system be inconsistent?
\end{enumerate}

{\bf Solution}\\
\ \    

\newpage
\section{$< 10pts >$}
Assume that the row echelon form for the matrix $\bf A$ involves no free
variables.
Answer {\bf True} or {\bf False} (with brief justification!):
\begin{enumerate}
\item The system ${\bf A}{\bf x} = {\bf b}$ will have infinitely
many solutions.
\item The system ${\bf A}{\bf x} = {\bf b}$ will be inconsistent.
\item The system ${\bf A}{\bf x} = {\bf b}$ will be consistent.
\item {\bf A} is nonsingular.
\item The system ${\bf A}{\bf x} = {\bf 0}$ will have a unique
solution.
\end{enumerate}
{\bf Solution}\\
\ \    


\end{document}
