\documentclass[10pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
%\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
%\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}           % Good for US Letter paper

\topmargin=-0.75in
\textheight=9.5in
\usepackage{fancyhdr} 
\renewcommand{\headrulewidth}{0pt}  % Otherwise there's a rule under the header
\setlength{\headheight}{15.2pt}
\fancyhf{}
\pagestyle{fancy} 

\cfoot{Page \thepage {} of 2}
% \pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{flushright}
Name \underline{\hspace{60mm}} \\ 
$\,$ \\
Student Number \underline{\hspace{60mm}}
\end{flushright}   
\vspace{2mm}

\begin{center}   
{\Large \textbf{STA 431 Quiz 2}}\\
\vspace{1 mm}
\end{center}

\noindent
% \emph{Calculators are allowed.} 

%\vspace{3mm}

\begin{enumerate} 

\item (3 points) Let $\mathbf{A}$ be a real, symmetric, positive definite matrix. Show that the eigenvalues of $\mathbf{A}$ are all strictly positive. Start with the definition $\mathbf{Ax}=\lambda\mathbf{x}$.

\vspace{100mm}


\item (3 points) Although eigen\emph{vectors} are always non-zero, it is possible for an eigen\emph{value} to equal zero. Let $\mathbf{A}$ be a square matrix, not necessarily symmetric, and let $(\lambda,\mathbf{x})$ be an (eigenvalue, eigenvector) pair with $\lambda=0$. Show that $\mathbf{A}$ does not have an inverse. 

\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item (4 points) Let the $p \times 1$ random vector $\mathbf{x}$ have expected value $\boldsymbol{\mu}$ and variance-covariance matrix $\mathbf{\Sigma}$, and let $\mathbf{A}$ be an $m \times p$ matrix of constants. Using the definition of a variance-covariance matrix on the formula sheet and familiar properties of expected value, derive the variance-covariance matrix of $\mathbf{Ax}$. 



% 



\end{enumerate}


\end{document}