习题集生成器

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\lhead{MATH 221: Linear Algebra}
\rhead{Problem Set 3}
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\begin{document}

\begin{center}
  {\large MATH 221: Linear Algebra}\\[2pt]
  {\LARGE\bfseries Problem Set 3\ifsolutions\ --- Solutions\fi}\\[4pt]
  Prof. A. Example \quad\textbullet\quad
  Due Friday, October 10 \quad\textbullet\quad
  Total: 20 points
\end{center}

\noindent Name: \rule{6cm}{0.4pt} \hfill Collaborators: \rule{5cm}{0.4pt}\par

\noindent\textit{Show all your work. You may discuss the problems, but write up your own solutions.}\par

\problem{1}{Eigenvalues}{10 points}
Let $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$.
\begin{enumerate}[label=(\alph*)]
  \item \textit{(4 points)} Find the characteristic polynomial of $A$.
\begin{solution}
  $\det(A - \lambda I) = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3$.
\end{solution}
  \item \textit{(6 points)} Find the eigenvalues and an eigenvector for each.
\begin{solution}
  $\lambda_1 = 1$ with $v_1 = (1,-1)^T$, and $\lambda_2 = 3$ with $v_2 = (1,1)^T$.
\end{solution}
\end{enumerate}

\problem{2}{Rank}{4 points}
Which matrix has rank 2?
\begin{enumerate}[label=(\Alph*)]
  \item $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$
  \item $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$
  \item $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
\end{enumerate}
\begin{solution}
  Answer: (B).
  Its determinant is $-2 \neq 0$.
\end{solution}

\problem{3}{Proof}{6 points}
Show that if $A$ is invertible, then $0$ is not an eigenvalue of $A$.
\begin{solution}
  If $Av = 0$ for some $v \neq 0$, then $v = A^{-1}Av = 0$, a contradiction.
\end{solution}

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