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\lhead{{\sf Summer Bridge Course (Algebra)}: Monday July 27}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesday, July 28 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Eigenvalues and eigenvectors. \\

\noindent \textbf{Do}: Answer the following questions. \\
%%------------------------------------------------------------------------

%% HW 15, Problem 1
\begin{exercise}
  Find the eigenvalues and eigenvectors for the following matrices:
  \[
  \A=\begin{bmatrix}-1&0&1&0\\2&1&2&1\\0&0&-1&0\\4&0&-6&1\end{bmatrix}
  \,,\qquad\quad
  \B=\begin{bmatrix}1&0&0&1\\2&1&0&-4\\1&0&1&-2\\0&0&0&1\end{bmatrix}
  \,,\qquad\quad
  \bold{J}_\lambda=\begin{bmatrix}\lambda&1&&\\&\lambda&\ddots& \\ &&\ddots&1\\ &&&\lambda\end{bmatrix}.
  \]
\end{exercise}

\medskip

%\begin{solution}
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%\end{solution}

%%------------------------------------------------------------------------

%% HW 15, Problem 2
\begin{exercise}
  The \emph{characteristic polynomial} of \(\A\) is
  \(\chi_{\A}(t)=\det(t\I-\A)\). Suppose this factors as
  \[
  \chi_{\A}(t)=(t-\lambda_1)(t-\lambda_2)\cdots
  (t-\lambda_n).
  \]
  \begin{enumerate}[label=(\roman*)]
    \item Plug in \(t=0\) and find a formula for \(\det\A\) in terms
      of the eigenvalues of \(\A\). 
    \item The \emph{trace} of \(\A\), denoted \(\tr\A\), is the sum of the
      diagonal entries, which is also equal to the sum of the
      eigenvalues. If \(\A\) is \(2\times 2\), then
      \[
      \A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\mbox{\quad has\quad}
      \det(tI-\A)=t^2-(a+d)t+(ad-bc).
      \]
      Write a formula for the characteristic polynomial of a \(2\times 2\)
      matrix in terms of \(\det\A\) and \(\tr\A\). 
    \item Suppose \(\A\) is an \(n\times n\) matrix with characteristic
      polynomial \(\chi_{\A}(t)=t^n+c_{n-1}t^{n-1}+\cdots+
      c_1t+c_0\). Describe \(\det\A\) and \(\tr\A\) in terms of the
      \(c_i\)'s.
    \item Explain why \(\A\B-\B\A=\I\) is impossible for \(n\times n\) matrices. \\
  \end{enumerate}  
\end{exercise}

%\begin{solution}
%\end{solution}

\medskip

 %%------------------------------------------------------------------------

%% HW 15, Problem 3
\begin{exercise}
  Suppose \(\A\) is a \(3\times 3\) matrix with eigenvalues \(0\), \(3\), and \(5\), with respective eigenvectors \(\u\), \(\v\), and \(\w\).
  \begin{enumerate}[label=(\roman*)]
  \item Give a basis for the nullspace and a basis for the column space. 
  \item Find a particular solution to \(\A\x=\v+\w\). Then, find all solutions. 
  \item Explain why \(\A\x=\u\) has no solution. 
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

 %%------------------------------------------------------------------------ 

\end{document}
