\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\bx{\boldsymbol{x}}
\def\by{\boldsymbol{y}}
\def\ba{\boldsymbol{a}}
\def\be{\boldsymbol{e}}
\def\A{{\boldsymbol A}}
\def\B{{\boldsymbol B}}
\def\I{{\boldsymbol I}}

\begin{document}

\title{Finite Mathematics}
\author{Homework 3}
\date{Due in class Monday, March 2, 2020}
\maketitle

\emph{Note:} Assignments marked (*) are \emph{not} for bonus credit.
They will not be graded.  Do not turn them in.  However, they will be
discussed in the tutorial and typically example solutions are
available online (in case of Hefferon's book, there is a PDF with
solutions available on the book's web site.)

\begin{enumerate}

\item *Demonstrate that
\[
  \A =
  \begin{pmatrix}
    2 & 2 & 3 \\ 1 & 0 & 1 \\ 1 & 1 & 1 
  \end{pmatrix}
  \quad \text{and} \quad
  \B =
  \begin{pmatrix}
    -1 & 1 & 2 \\
    0 & -1 & 1 \\
    1 & 0 & -2
  \end{pmatrix}
\]
are both nonsingular by showing that $\A \B = \I$.

\item *Give an example that shows, in general, $\A \B \neq \B \A$.

\item Show that if $\A$ is symmetric and nonsingular, then $\A^{-1}$
is symmetric.
\hfill(2)

\item *Hefferon, p.\ 252, Exercises 4.15

\item Compute the inverse, if possible, of each of the following
matrices.

\begin{enumerate}
\item $\displaystyle \A =
\begin{pmatrix}
  1 & 4 \\ -2 & 3
\end{pmatrix}$

\item $\displaystyle \A =
\begin{pmatrix}
  1 & 2 & 3 \\
  2 & 4 & 5 \\
  3 & 5 & 6 
\end{pmatrix}$

\item $\displaystyle \A =
\begin{pmatrix}
  1 & 2 & 1 \\
  -2 & 1 & 8 \\
  1 & -2 & -7
\end{pmatrix}$
\end{enumerate}
\hfill(2+2+2)

\item VMLS, p.\ 218, Exercise 11.9.
\hfill(2)
\end{enumerate}
\end{document}
