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\homework{13: Linear Algebra}{Due Friday, Week 7}

Solve \textbf{one} of the following \textbf{three} problems!

\begin{enumerate}
\item Prove the following claims about permutation matrices:
\begin{enumerate}
\item If $P$ is a permutation matrix with associated permutation $\sigma$, then $P^{-1}$ is a permutation matrix with associated permutation $\sigma^{-1}$.
\item If $P$ is a $n \times n$ permutation matrix with associated permutation $\sigma$, then $(v_1, \ldots v_n) \cdot P = (v_{\sigma^{-1}(1)}, \ldots v_{\sigma^{-1}(n)})$.
\end{enumerate}
\item Consider the Pell sequence $\{p_i\}_{i=1}^\infty$, defined recursively as follows:
\begin{itemize}
\item $p_0 = 0$.
\item $p_1 = 1$.
\item $p_n = 2p_{n-1} + p_{n-2}$.
\end{itemize}
The first ten Pell numbers are listed here:
\begin{align*}
0, 1, 2, 5, 12, 29, 70, 169, 408, 985, \ldots
\end{align*}
\begin{enumerate}
\item Find a matrix $P = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$ such that
\begin{align*}
\begin{bmatrix} a & b \\ c & d\end{bmatrix} \cdot \begin{bmatrix} p_n \\ p_{n-1} \end{bmatrix} =  \begin{bmatrix} p_{n+1} \\ p_{n} \end{bmatrix} .
\end{align*}
\item Find the eigenvalues and corresponding eigenvectors of $P$.
\item Use this information along with the methods we discussed in class to find $p_{50}$.
\end{enumerate}
\item Consider the following two-player game that you can play on a $n \times n$ grid:
\begin{itemize}
\item There are two players, 1 and 2.  
\item These two players alternate putting real numbers into the entries of the matrix.
\item Once the matrix is filled, player 1 wins if the determinant of this matrix is nonzero; player 2 wins if the determinant is zero.
\end{itemize}
Can you find a strategy for one of these players that guarantees that they'll win? 
\end{enumerate}
\end{document}






