\documentclass[12pt]{amsart}

\setlength{\parindent}{0pt}
\setlength{\textheight}{9in}
\setlength{\voffset}{-1in}
\setlength{\textwidth}{6.5in}
\setlength{\hoffset}{-.75in}
\everymath={\displaystyle}
\pagestyle{empty}

\newcommand{\R}{\ensuremath{\mathbb{R}}}

\begin{document}

\thispagestyle{empty}

\textbf{Math 228: Intro to Lin Alg \& Diff Eqns}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Quiz 8}\quad \textsl{April 4, 2002}
\hfill\makebox[2.5in][r]{Section:\hrulefill}

\vspace{.5in}

1. Mark each of the following statements as either true or false:

\vspace{.25in}

a) An $n\times n$ symmetric matrix always has $n$ different eigenvalues.

\vspace{.25in}

b) An $n\times n$ symmetric matrix always has $n$ linearly independent
eigenvectors.

\vspace{.25in}

c) Two different eigenvectors of a symmetric matrix are always orthogonal.

\vspace{.25in}

d) A symmetric matrix can always be diagonalized.

\vspace{.75in}


2.  The matrix $A = \left[\begin{array}{rrr}
4&2&0\\
2&1&0\\
0&0&3\end{array}\right]$ has eigenvalues $\lambda_1=0,\lambda_2=3,
\lambda_3=5$.
Find an orthogonal matrix $Q$ so that 
$Q^T\!\!AQ = \left[\begin{array}{rrr}
0&0&0\\
0&3&0\\
0&0&5\end{array}\right]$

\vfill


\end{document}

