%LaTeX file
\documentclass[12pt]{article}
\usepackage[width=7.25in,height=10in]{geometry}
\usepackage{graphicx}
\newcommand{\eqref}[1]{(\ref{#1})}
\pagestyle{empty}
\begin{document}
\begin{tabbing}
\hskip36pc \= Math.\ 362-01\\
\>Summer, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
{\bf Fourth Exam} \\
{\em Thursday, July 28, 2005}
\end{center}

\noindent{\small This exam is closed book. Make sure your name is on
all pages.  You should put your work on your own paper, and you may
keep this exam sheet upon leaving.  Be sure to check your work
carefully, and to show intermediate computations in a logical
presentation.  \textbf{Full credit will not be given unless
computations are shown.} Each entire problem is worth 50 points.}

\Large
\begin{enumerate}
   \item Suppose
            $$A = \left(\begin{array}{rr}
                  4 & -1 \\ -1 & 4
            \end{array}\right).$$

   \begin{enumerate}
      \item Compute a set of eigenvalues and eigenvectors of $A$.
       \item Is this matrix $A$ diagonalizable?  If so, then write
            down a matrix $V$ and a diagonal matrix $D$ so that $A =
            V D V^{-1}$.
   \end{enumerate}
   \item Suppose
            $$A = \left(\begin{array}{rr}
                  3 & 2 \\ 4 & 4
            \end{array}\right).$$
         In the following computations, you may use decimal
         approximations.
         \begin{enumerate}
            \item Compute a QR factorization of $A$ by using
                  elementary rotations.
            \item Using your QR factorization, find the solution $x$ to the
                  system
                  $Ax=b$, where $b = (-1,1)^t$.
         \end{enumerate}
\end{enumerate}

\end{document}
