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\begin{document}
\title{314 '07-QUIZ 2}
\author{Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }
\maketitle
\section{$< 10pts >$}
Find all values of $c$ for which the matrix ${\bf A}$ is singular, with
\[ {\bf A} = \left(
\begin{array}{rrr}
1-c & -2 & 2 \\0 & 2-c & 3\\0 & 0 & 3-c
\end{array}\right)   \]
{\bf Solution}\\
\ \    

\newpage
\section{$< 10pts >$}
A top secret message was encoded into the string
%\[ -8,7,-7,-1,12,-14,26,15,-45 \]
%\[ -1,8,-8,37,23,-42,19,20,-34 \]
\[ -1, 12, -14, -4, 15, -15, 9, 18, -30, 5, 5, -35 \]
using the matrix
\[ {\bf A} = \left(
\begin{array}{rrr}
1 & -1 & 1 \\1 & 0 & 0 \\-1 & 0 & -1 
\end{array}\right)   \]
and the correspondence:\\
$\mbox{space}=0$,$A=1$,$B=2$,$\ldots$,$Z=26$,$(\mbox{.\ ,\ ?\ !\ :\ *})=
(27,\ldots,32)$. What was the message?

{\bf Solution}\\
\ \
                                                                                
                                                                                
                                                                                

\newpage
\section{$< 10pts >$}
Consider the matrix
\[ {\bf A} = \left(
\begin{array}{rrrr}
1 & 2 & -1 & 1 \\-1 & -3 & 0 & -1\\1 & 1 & -1 & 0 \\1 & 0 & -2 & 2
\end{array}\right)   \]
\begin{enumerate}
\item
Use Gauss elimination to find the determinant $det{\bf A}$ 
\item
What is the determinant of the inverse, $det{\bf A}^{-1}$? 
%\item
%Give the entry $(1,2)$ of the inverse.
\end{enumerate}

{\bf Solution}\\
\ \ 



\newpage
\section{$< 10pts >$}
Use Cramer's rule to solve the system
\[
\begin{array}{rcrcrcr}
 x_1 & - & x_2 &+& x_3 &=& 1 \\
-x_1 & + &2x_2 &-& x_3 &=& 0 \\
2x_1 & - & x_2 &+&3x_3 &=& 0
\end{array}
\]
{\bf Solution}\\
\ \    


\end{document}
