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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 25 October 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Calculate each of the following determinants:
\begin{enumerate}
\item $\det\left(\left[\begin{array}{cc}2 & 3 \\ -1 & 1\end{array}\right]\right)$
\item $\det\left(\left[\begin{array}{ccc} 1 & 2 & -1\\ 2 & 1 & 0\\ 0 & 1 & 0\end{array}\right]\right)$
\end{enumerate}

\item Prove that the determinant of an upper triangular matrix is equal to the product of the diagonal entries of the matrix.

\item Complete problems 12.14.2, 12.14.3, 12.14.9 on pages 483-485 in Coding the Matrix.

\item Suppose that $\lambda$ is an eigenvalue of the invertible matrix $A$ having a corresponding eigenvector $\vec{v}$. Prove that $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$. What is a corresponding eigenvector?

\item Use the previous problem to prove that a square matrix $A$ is invertible if and only if $0$ is not an eigenvalue of $A$.

\item Complete the problem set found at \url{autolab.andrew.cmu.edu}. The submission for this is directly on autolab, no need to hand it in on paper.
\end{enumerate}
\end{document}