\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}}

\begin{document}

\title{Finite Mathematics}
\author{Homework 2}
\date{Due in class Wednesday, February 19, 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 *Given the matrices
\begin{gather*}
  \A =
  \begin{pmatrix}
    2 & 2 \\ -1 & 3
  \end{pmatrix} \,,
  \quad
  \B =
  \begin{pmatrix}
    -1 & 1 & -2 \\ 1 & 3 & 4
  \end{pmatrix} \,,
  \quad
  \bx =
  \begin{pmatrix}
    2 \\ 3
  \end{pmatrix} \,,
  \quad
  \by = 
  \begin{pmatrix}
    1 & 2 & -1
  \end{pmatrix} \,,    
\end{gather*}
compute the following.  Note that not all operations may be
well-defined.
\begin{enumerate}
\item $\A + \B$
\item $\A \B$
\item $\B \A$
\item $\A \bx$
\item $\B \bx$
\item $\B^T \A$
\item $\B^T \A^T$
\item $\by \B^T$
\item $\bx^T \A \bx$
\item $\bx \B^T \by^T$
\end{enumerate}


\item Prove the following:
\begin{enumerate}
\item If $\A$ and $\B$ are symmetric, then $\A \B$ is symmetric if and
only if $\A \B = \B \A$.  (We then say that ``$\A$ and $\B$
commute''.)

\item $\A \A^T$ is symmetric.

\item If $\A$ is a square matrix, then $\A + \A^T$ is symmetric.
\end{enumerate}
\hfill(3)

\item *Hefferon, p.\ 54, Exercises 1.12

\item If there exist solutions of the homogeneous system of linear
equations other than $\bx=\boldsymbol 0$, express the general solution
as a linear combination of linearly independent column vectors:
\begin{equation*}
\begin{aligned}[t]
  &x_1 - x_3 = 0 \\
  &3 \, x_1 + x_2 + x_3 = 0 \\
  &-x_1 + x_2 + 2 \, x_3 = 0
\end{aligned}
\end{equation*}
\hfill(2)

\item Find the general solution of the given set of equations, or else
show that there is no solution.
\begin{equation*}
% \item $\displaystyle
% \begin{aligned}[t]
%   & 2 \, x_1 + x_2 + x_3 = 2 \\
%   & - x_1 + x_3 = 1 \\
%   & x_1 + x_2 + 2 \, x_3 = 3
% \end{aligned}$
% \item $\displaystyle
% \begin{aligned}[t]
%   & 2 \, x_1 + x_2 + x_3 = 0 \\
%   & - x_1 + x_3 = -1 \\
%   & x_1 + x_2 + 2 \, x_3 = 1
% \end{aligned}$
\begin{aligned}[t]
  & x_1 - x_2 + x_3 + x_4 = -1 \\
  &x_2 + x_3 +3 \,  x_4 = 2 \\
  &x_1 + 2 \, x_3 + 4 \, x_4 = 1 \\
  & x_2 +  x_3 + 3 \, x_4 = 2
\end{aligned}
\end{equation*}
\hfill(3)

\item (Lial \emph{et al.}, p.\ 74, Question 49) The U-Drive
Rent-A-Truck Co.\ plans to spend \$6 million on 200 new vehicles.
Each van will cost \$20,000, each small truck \$30,000, and each large
truck \$50,000. Past experience shows that they need twice as many
vans as small trucks. How many of each kind of vehicle can they buy?
\hfill(2)

\end{enumerate}

\end{document}
