
\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer I 2012 \\
       Homework 1}

\begin{document}
  \maketitle

  \begin{enumerate}
    \item  Solve the following system of equations using Gaussian elimination.  [5]
      \begin{align*}
        x + 2y + 3z & = 0 \\
        4x + 5y  + 6z & = 2 \\
        7x + 8y + 9z & = 4
      \end{align*}
      You must give a complete description of the set of solutions.
    \item  For which real values of $a$ and $b$ does the following system have a unique solution?  or infinitely many solutions?  or no solutions?  [10]
      \begin{align*}
        x - 2y & = 1 \\
        ax + y & = b \\
      \end{align*}
    \item  Let $X = \set{(a,b)\in\RR^2}{a = b\lor a = -b}$.  Show that there is no system of linear equations whose solution set is exactly $X$.  [10]
    \item  Prove that a swap operation can be implemented by a sequence of row combination and scaling operations.  [10]
    \item  Give a complete description of all $2\times 2$ matrices in reduced row echelon form.  \emph{Or}: Give a complete description of all $2\times 2$ matrices in row echelon form.  [10]
    \item  Prove or disprove: $A^2 = 0$ implies $A = 0$ for all $2\times 2$ matrices $A$.  [15]

  \end{enumerate}
\end{document}
