\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 5}
\date{Due in class Tuesday, March 12, 2019}
\maketitle

\begin{enumerate}

\item Compute the determinant and 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}

\item Find the general solution of the given set of equations, or else
show that there is no solution.
\begin{enumerate}
\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 $\displaystyle
\begin{aligned}[t]
  & 2 \, x_1 + x_2 + x_3 = 0 \\
  & - x_1 + x_3 = -1 \\
  & x_1 + x_2 + 2 \, x_3 = 1
\end{aligned}$

\item $\displaystyle
\begin{aligned}[t]
  & x_1 - x_2 + x_3 + x_4 = -1 \\
  &x_2 + x_3 +3 \,  x_4 = 2 \\
  &x_1 + 2 \, x_3 + 4 \, x_4 = 1 \\
  & x_2 +  x_3 + 3 \, x_4 = 2
\end{aligned}$

\end{enumerate}

\item Find all eigenvalues and eigenvectors for each of the following
matrices.  If possible, find a matrix $\boldsymbol S$ and a diagonal
matrix $\boldsymbol D$ such that
\[
  \A = \boldsymbol S \boldsymbol D \boldsymbol S^{-1} \,.
\]

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

\item $\displaystyle \A =
\begin{pmatrix}
  1 & 0 & 0 \\
  2 & 1 & -2 \\
  3 & 2 & 1 
\end{pmatrix}$

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

\end{enumerate}

\end{enumerate}
\end{document}
