%---*-LaTeX-*-------

\documentclass{article}
\usepackage{amsmath}

\input{ExamMacros.tex}

\prof{}
\class{Math 221}
\quizdate{}
%\examname{Practice Problems}

\begin{document}
\centerline{\large Practice Problems for Exam 1 -- Math 221}
\bigskip\bigskip
%\pagestyle{empty}

%\heading

The exam will cover material {\em through} \S 2.3.  The homework
should be reasonably indicative of the types of problems you may be
expected to solve on the exam.  Also, you should be able to solve the
additional problems listed below.

\begin{problem}
Suppose that $C$ is a $3 \times 3$ matrices.  Let $E$ be the appropriate
product of elementary matrices that ``puts C is reduced row echolon from'';
 in this case assume $E
= \begin{bmatrix} 1 & 0 & -2 \\ 0 & 0 & 1 \\ 3 & -1 & 0 \\
\end{bmatrix}$ and $E C =
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}.  $
Form the new $3 \times 5$ matrix 
        $A = \begin{bmatrix} 
                C &
                \begin{matrix} 2 \\ -1 \\ 0 \end{matrix} & 
                \begin{matrix} 1 \\ 0 \\ 10 \end{matrix} 
             \end{bmatrix}$, 
and solve the
system of equations $A \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5
\end{bmatrix} = 0$.  Be sure to give all of the solutions.
\end{problem}

\begin{problem}
Assume that the trigonometric function $f$ has the following form:
\[ f(\theta) = a \sin(\theta) + b \cos \left (\frac{\theta}{2} \right ) + c \]
Write down the system of linear equation that $a$, $b$ and $c$ must
satisfy in order that the graph of $f$ pass through the points $p$ and
$q$ where $p$ and $q$ are given by:
\[ p = (\pi,0) \text{  and  } q = (3\pi/2,0) \]

\noindent Find all of the possible functions $f$ of the given form can one find that pass
through $p$ and $q$.
\end{problem}
%
%


\begin{problem}
Decide whether each of the following statement is true or false, and
give a brief argument.  Remember, to prove that a statement is {\em
false}, it is usually necessary to provide an example.

\begin{subproblem}
        If $A$ and $B$ are $n \times n$ matrices, 
        $\det(A + B) = \det(A) + \det(B)$.
\end{subproblem}

\begin{subproblem}
        If $A$ and $B$ are $n \times n$ matrices,
        $AB = BA$.
\end{subproblem}

\begin{subproblem}
        If $A$ is a symmetric $n \times n$ matrix, then $A^T$ is
        also symmetric.
\end{subproblem}

\begin{subproblem}
        An $n \times n$ matrix $A$ is invertible if and only if
        $\det (E A) \not = 0$ for any elementary matrix $E$.
\end{subproblem}

\begin{subproblem}
        If $A$ and $B$ are symmetric matrices, then $AB$ is a
        symmetric matrix.
\end{subproblem}

\end{problem}

\begin{problem}
        Refer to the following problems from the text.
\obeylines
\S 1.1: 7
\S 1.2: 8, 12, 17, 21, 25
\S 1.3: 12
\S 1.4: 6
\S 1.5: 14
\S 1.6: 15, 17, 25
\S 1.7: 15, 19
\S 2.1: 17,13
\S 2.3: 12, 14, 15, 18

Many of these may have already been assigned.
\end{problem}

\end{document}
