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

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\x{\boldsymbol{x}}

\begin{document}

\title{Applied Differential Equations and Modeling}
\author{Homework 4}
\date{Due in class Tuesday, March 13, 2018}
\maketitle

\begin{enumerate}

\item Find the eigenvalues and eigenvectors for each of the following
matrices. 

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

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

\end{enumerate}

\item For the matrices from Problem 1, sketch the phase portrait for
the corresponding linear differential equation $\dot \x = A\x$.

\item Describe how the nature of the eigenvalues depends on the
parameter $\alpha$ in the matrix
\[
  A =
\begin{pmatrix}
  2 & \alpha \\ 1 & -3
\end{pmatrix} \,.
\]

\item Verify that the inverse of the matrix 
\[
  A =
\begin{pmatrix}
  a_{11} & a_{12} \\ a_{21} & a_{22}
\end{pmatrix} 
\]
is given by the formula
\[
  A^{-1} = \frac1{\det A} \,
  \begin{pmatrix}
     a_{22} & -a_{12} \\ -a_{21} & a_{11}
  \end{pmatrix} \,.
\]

\item Find the solution to the linear equation
\[
  \dot \x =
  \begin{pmatrix}
    2 & -1 \\ 3 & -2 
  \end{pmatrix}
  \x \,, \qquad
  \x(0) =
  \begin{pmatrix}
    2 \\ 5
  \end{pmatrix} \,.
\]

\item Suppose that $\x_1(t)$ and $\x_2 (t)$ are solutions of
$\dot \x = A \x$.  Show that
$\x(t) = \alpha \, \x_1(t) + \beta \, \x_2(t)$, where $\alpha$ and
$\beta$ are arbitrary constants, is also a solution.




\end{enumerate}
\end{document}
