\documentclass{amsart}


\title{Matrix theory autumn 2011: homework 2}

\author{James Cummings}


\begin{document}

\maketitle
    
   Homework due at start of class on Wed 14 September. 



   


\section{Matrices and vectors}

\begin{enumerate}

\item  Let $A$ be an $n \times n$ matrix. The {\em trace} of $A$ (denoted by $tr(A)$) is defined to be the sum of the
    diagonal elements of $A$, that is $\sum_{i=1}^n a_{ii}$. Prove that for any two $n \times n$ matrices
    $A$ and $B$, $tr(A B) = tr(B A)$. Is it true for any three such matrices $A, B, C$ 
    we have $tr(A B C) = tr(B A C)$? 




\item   Let $A$ be an $n \times n$ matrix where every entry $a_{ij}$ is either $0$ or $1$. 
        Prove  that for $t > 0$ the $(i, j)$-entry in the matrix power
         $A^t$ (the product of $t$ copies of $A$) is equal to the number
        of sequences $(x_0, \ldots x_t)$ such that
\begin{enumerate}
\item  Each $x_k$ for $0 \le k \le t$  is a natural number with $1 \le x_k \le n$.
\item  $x_0 = i$ and $x_t = j$.
\item  For all $k$ with $0 \le k < t$, $a_{x_k x_{k+1}} = 1$.
\end{enumerate}

\item  Let $X$ be the set of all real $2 \times 2$  matrices which are of the special form
\[
\left (
\begin{array}{rr}
a & - b \\
b &   a \\
\end{array}
\right )
\] for some real numbers $a$ and $b$.
  Let $I$ be the $2 \times 2$ identity matrix, that is
\[I = 
\left (
\begin{array}{rr}
1 & 0 \\
0 & 1 \\
\end{array}
\right )
\]
  
\begin{enumerate}
\item    Prove that $X$ is closed under matrix addition and multiplication.
\item    Prove that $A B = B A$ for all $A, B \in X$.
\item    Prove that if $A \in X$ and $A$ is not the zero matrix, there is $B \in X$ such that $A B = I$.
\item    Prove that there is $A \in X$ such that $A^2 = -I$. 
\end{enumerate}


\item  Let 
 \[
A=
\left (
\begin{array}{rr}
a & b \\
c & d \\
\end{array}
\right )
\]
 be a $2 \times 2$ real matrix. Prove that if $a d - b c < 0$ there is no matrix $B$ such that
     $A = B^2$. Is it the case that whenever $a d - b c \ge 0$ then such a matrix $B$ exists? 


\end{enumerate} 

\section{Mathematica} 


    This section is {\em not for credit}, {\em you may collaborate} and {\em you need not hand anything in.}
    But you still should do it because probably none of this  will be true of the computer algebra sections in later
    homeworks. 
 


\begin{enumerate}

\item  Install the student version of Mathematica (for which CMU really and truly does 
       have a license) on your personal computer. 

\item  Start Mathematica.

\item  Work through the first couple of sections of the online tutorial
       at the URL

\begin{verbatim}
http://reference.wolfram.com/mathematica/tutorial/GettingStartedOverview.html
\end{verbatim}

\item  Read enough of 

\begin{verbatim}
http://reference.wolfram.com/mathematica/howto/CreateAMatrix.html
\end{verbatim}

\noindent and

\begin{verbatim}
http://reference.wolfram.com/mathematica/guide/MatrixOperations.html
\end{verbatim}

\noindent   to figure out how to enter a matrix and do elementary matrix arithmetic.

\item  Working in Mathematica, create matrices 
\[
A = \left (
\begin{array}{cc}
1 & 2 \\
4 & 5 \\
\end{array}
\right ),
B = \left (
\begin{array}{cc}
3 & 7 \\
6 & 8 \\
\end{array}
\right ).
\]

\item   Use Mathematica to compute $A + B$, $A B$, $B A$,  $A^{100}$.
  Check the first three answers by hand. 


\item Export your results to a PDF file. 

\end{enumerate}





\end{document}



