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\begin{document}


\section{Ma 227\qquad \qquad \qquad \qquad \qquad \qquad Practice for Exam I
\qquad \qquad \qquad \qquad }

$\qquad \qquad \qquad \qquad \qquad $

\subsection{\protect\vspace{1pt}Problem 1 \ }

Find the eigenvalues and eigenvectors of

\vspace{1pt}

\begin{center}
\qquad \qquad \qquad \qquad \qquad $\left[ 
\begin{array}{lll}
1 & 3 & -2 \\ 
0 & 2 & 4 \\ 
0 & 1 & -1%
\end{array}%
\right] \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad $
\end{center}

\vspace{1pt}

\subsection{Problem 2}

\vspace{1pt}a) \ Row reduce the matrix $B$ to echelon form, where

\qquad \qquad

\begin{center}
$\vspace{1pt}B=\left[ 
\begin{array}{llll}
2 & 1 & 1 & 2 \\ 
1 & 2 & 1 & 1 \\ 
-1 & 1 & -2 & -1%
\end{array}%
\right] $ .
\end{center}

\vspace{1pt}\qquad \qquad \qquad \qquad 

\vspace{1pt}\vspace{1pt}b) \ Find all values of $t$\ for which the system of
equations below will have a non-trivial solution.

\qquad \qquad \qquad 
\begin{eqnarray*}
x-ty+z &=&0 \\
2y+tz &=&0 \\
y+z &=&0
\end{eqnarray*}

c) Solve, if possible, the system

\begin{enumerate}
\item 
\begin{eqnarray*}
x_{1}+2x_{2}-3x_{3}-4x_{4} &=&6 \\
x_{1}+3x_{2}+x_{3}-2x_{4} &=&4 \\
2x_{1}+5x_{2}-2x_{3}-5x_{4} &=&10\smallskip 
\end{eqnarray*}
\end{enumerate}

\subsection{\protect\vspace{1pt}Problem 3}

Let

\vspace{1pt}\qquad \qquad \qquad 
\[
B=\left[ 
\begin{array}{lll}
1 & 2 & 1 \\ 
1 & 3 & 2 \\ 
1 & 0 & 1%
\end{array}%
\right] 
\]%
Find $B^{-1}$.\qquad 

\vspace{1pt}

\subsection{Problem 4 \ }

a) Find the general solution of 
\[
x^{\prime }=Ax
\]

where 
\[
A=\left[ 
\begin{array}{cc}
1 & -1 \\ 
-2 & 0%
\end{array}%
\right] 
\]

b) Write the equation $y^{\prime \prime \prime }-2y^{\prime \prime
}+2y^{\prime }+4y=\sin t$ in \underline{normal} form $(x^{\prime }=Ax+f)$

\vspace{1pt}

c) Solve the initial value problem%
\[
x^{\prime }\left( t\right) =\left[ 
\begin{array}{ccc}
1 & 2 & -1 \\ 
1 & 0 & 1 \\ 
4 & -4 & 5%
\end{array}%
\right] \text{ \ \ \ \ \ }x\left( 0\right) =\left[ 
\begin{array}{c}
-1 \\ 
0 \\ 
0%
\end{array}%
\right] 
\]

\vspace{1pt}

\qquad \qquad \qquad $\ \ \ \ \ \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \ \ \ \ \ \ $

\qquad \qquad \qquad \qquad $\ $

\vspace{1pt}

\end{document}
