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\lhead{{\sf Bridge course (Algebra)}: Thursday July 24}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, July 25 2025}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

%$\;$ \vspace{-4mm}

\noindent \textbf{Topic}: Eigenvalues. \\

\noindent \textbf{Do}: Answer the following questions. Assume that all matrices are over the field $K=\mathbb{C}$.

\begin{enumerate}

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\item Find the eigenvalues and eigenvectors for the following matrices:
  \[
  \A=\begin{bmatrix}-1&0&1&0\\2&1&2&1\\0&0&-1&0\\4&0&-6&1\end{bmatrix}
  \,,\qquad\quad
  \B=\begin{bmatrix}1&0&0&1\\2&1&0&-4\\1&0&1&-2\\0&0&0&1\end{bmatrix}
  \,,\qquad\quad
  \bold{J}_\lambda=\begin{bmatrix}\lambda&1&&\\&\lambda&\ddots& \\ &&\ddots&1\\ &&&\lambda\end{bmatrix}\,.
  \] 
  
%%------------------------------------------------------------------------
 
\item 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}
    \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
      \[
      \A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\mbox{\quad has\quad}
      \det(tI-\A)=t^2-(a+d)t+(ad-bc)=0.
      \]
      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}  

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\item Suppose $\A$ is a $3\times 3$ matrix with eigenvalues $0$, $3$, and $5$, with respective eigenvectors $\u$, $\v$, and $\w$.
  \begin{enumerate}
  \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}

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\end{enumerate}

\end{document}
