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\documentclass{article}
\usepackage{amsmath,amssymb}

\input{ExamMacros.tex}

%\prof{George McNinch}
\prof{Peter Cholak}
\class{Math 221}
\quizdate{}
%\examname{Practice Problems}

\begin{document}
\heading 

Here are some practice problems from Chapter 7.  The exam will
have at least one problem from sections 7.1 and 7.2.

Our exam will be in this room on Wednesday, December 18 from 8
am.\ to 10 am.\\

\begin{problem}
  Consider the $3 \times 3$ matrix 
  $A = \begin{bmatrix} 
    1 & 2 & -1 \\
    0 & 2 & -2 \\
    0 & 0 & -1 
  \end{bmatrix}$.
  \begin{subproblem}
    Find the eigenvalues of $A$.
  \end{subproblem}
  \begin{subproblem}
    The matrix $A$ is diagonalizable.  Why?
  \end{subproblem}
  \begin{subproblem}
    Compute $P$ so that $P^{-1} A P$ is a diagonal matrix.  Find this
    diagonal matrix.
  \end{subproblem}
  \begin{subproblem}
    Compute $A^5$ (do this the ``easy way'' using the matrix $P$
    from the previous part of the problem). (Hint: use elementary
    operations to find $A^{-1}$ and to multiply the matrices.)
  \end{subproblem}
\end{problem} 

\begin{problem}
 Let $S = 
 \begin{bmatrix}
   1 & 1 & 0 \\
   1 & 1 & 0 \\
   0 & 0 & 1 \\
 \end{bmatrix}$.
 \begin{subproblem}
   Find the eigenvalues of $S$.
 \end{subproblem}
 \begin{subproblem}
   Find 3 linearly independent eigenvectors for $S$.
 \end{subproblem}
 \begin{subproblem}
   Normalize these eigenvectors.
 \end{subproblem}
 \begin{subproblem}
   Find $P$ so that $P^{-1} S P$ is diagonal.  Show that
   you can choose $P$ to be an {\em orthogonal} matrix.
 \end{subproblem}
   \begin{subproblem}
     Compute $A^{6}$ (do this the ``easy way'' using the matrix
     $P$ from the previous part of the problem). Note since $P$
     is orthogonal $P^{-1} = P^{T}$ so no computation is needed
     to find $P^{-1}$. Why? And again you can use elementary
     operations to multiply the matrices.)
  \end{subproblem}
\end{problem}


\begin{problem}
  Determine whether the following matrixes are diagonalizable or
  not. Explain why or why not! Do not diagonalize!
  \begin{subproblem}
    $\begin{bmatrix}
    2 & 0 & 3 & 3\\
    0 & 2 & 3 & 3\\
    0 & 0 & 1 & -1\\
    0 & 0 & 0 & 1 \\
   \end{bmatrix}$.
  \end{subproblem}
  \begin{subproblem}
    $\begin{bmatrix}
    2 & 0 & 3 & 3\\
    0 & 2 & 3 & 3\\
    0 & 0 & 1 & -1\\
    0 & 0 & 0 & 3 \\
   \end{bmatrix}$.
  \end{subproblem}
 \begin{subproblem}
    $\begin{bmatrix}
    2 & 1 & 3 & 3\\
    0 & 2 & 3 & 3\\
    0 & 0 & 1 & -1\\
    0 & 0 & 0 & 3 \\
   \end{bmatrix}$.
  \end{subproblem}
\end{problem}

%\begin{problem}
%  (True/False and why...)  Suppose that the $n \times n$ matrix
%  $A$ has $n$ distinct eigenvalues $\{ \lambda_1, \lambda_2,
%  \dots, \lambda_n \}$. Then $\det(A) = \lambda_1  \lambda_2
%\cdots \lambda_n$.
%\end{problem}


%\begin{problem}
%  (Tell whether each is true or false and why) Suppose that the
%  characteristic polynomial of the $3 \times 3$ matrix $B$ is
%  $\det(\lambda I - B) = (\lambda - 1)^2 \lambda$.
%  \begin{subproblem}
%    The eigenspace corresponding to $\lambda = 1$ is 2 dimensional.
%  \end{subproblem}
%  \begin{subproblem}
%    The eigenspace corresponding to $\lambda = 0$ is 1 dimensional.
%  \end{subproblem}
%  \begin{subproblem}
%    The matrix $B$ is not invertible.
%  \end{subproblem}
%\end{problem}
\end{document}

