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

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\F{{\boldsymbol F}}
\def\v{{\boldsymbol v}}
\def\x{{\boldsymbol x}}
\def\y{{\boldsymbol y}}
\def\A{{\boldsymbol A}}
\def\B{{\boldsymbol B}}
\def\I{{\boldsymbol I}}
\def\rank{\operatorname{rank}}

\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 4}
\date{Due in class Tuesday, March 5, 2019}
\maketitle

\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
  \x =
  \begin{pmatrix}
    2 \\ 3
  \end{pmatrix} \,,
  \quad
  \y = 
  \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 $\A \x$
\item $\B \x$
\item $\B^T \A$
\item $\B^T \A^T$
\item $\y \B^T$
\item $\x^T \A \x$
\item $\x \B^T \y^T$
\end{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 Prove the following:
\begin{enumerate}
\item If $\A$ is symmetric and nonsingular, then $\A^{-1}$ is
symmetric.

\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}

\item In each case, reduce $\A$ to row echelon form and determine
$\rank \A$.

\begin{enumerate}
\item $\displaystyle \A =
\begin{pmatrix}
  1 & -3 & 4 \\ 2 & -6 & 8
\end{pmatrix}$

\item $\displaystyle \A =
\begin{pmatrix}
  0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 
\end{pmatrix}$

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

\item In each of the following, if there exist solutions of the
homogeneous system of linear equations other than $\x=\boldsymbol 0$,
express the general solution as a linear combination of linearly
independent column vectors.
\begin{enumerate}
\item $\displaystyle
\begin{aligned}[t]
  &x_1 - x_3 = 0 \\
  &3 \, x_1 + x_2 + x_3 = 0 \\
  &-x_1 + x_2 + 2 \, x_3 = 0
\end{aligned}$

\item $\displaystyle
\begin{aligned}[t]
  &x_1 - 2 \, x_2 + x_4 = 0 \\
  &2 \, x_1 + x_2 + x_3 - x_4 = 0 \\
  &x_1 + 2 \, x_2 + x_3 - 2 \, x_4 = 0 \\
  &3 \, x_1 + 3 \, x_2 + 2 \, x_3 - 3 \, x_4 = 0
\end{aligned}$

\end{enumerate}

\item Determine whether the members of the given set of vectors are
linearly independent.  If they are linearly dependent, find a linear
relation among them.

\begin{enumerate}
\item $\displaystyle
\v_1 =
\begin{pmatrix}
  1 \\ 1 \\ 0
\end{pmatrix} \,,
\quad
\v_2 =
\begin{pmatrix}
  0 \\ 1 \\ 1 
\end{pmatrix} \,,
\quad
\v_3 =
\begin{pmatrix}
  1 \\ 0 \\ 1
\end{pmatrix}$

\item $\displaystyle
\v_1 =
\begin{pmatrix}
  1 \\ 2 \\ -2
\end{pmatrix} \,,
\quad
\v_2 =
\begin{pmatrix}
  3 \\ 1 \\ 0
\end{pmatrix} \,,
\quad
\v_3 =
\begin{pmatrix}
  2 \\ -1 \\ 1
\end{pmatrix}\,,
\quad
\v_4 =
\begin{pmatrix}
  4 \\ 3 \\ -2
\end{pmatrix}$

\end{enumerate}

\end{enumerate}
\end{document}
