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\headers{Math343}{Homework 8}{Due Oct 15, 2025}

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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work..

\begin{enumerate}

\item True/False.  In the following, $A$, $B$, and $C$ are matrices for which the given expressions are defined.  Justify your answer.
\begin{enumerate}
	\item Each column of $AB$ is a linear combination of the columns of $B$ using weights from the corresponding column of $A$.
	\item The second row of $AB$ is the second row of $A$ multiplied on the right by $B$.
	\item $A^T + B^T = (A+B)^T$.
	\item $(AB)C = (AC)B$
	\item $(AB)^T = A^TB^T$
	\item If $A$ can be row reduced to the identity matrix, then $A$ must be invertible.
	\item Each elementary matrix is invertible.
	\item If $A$ is invertible, then the elementary row operations that reduce $A$ to the identity $I_n$ also reduce $A^{-1}$ to $I_n$.
	\item If the columns of an $n\times n$ matrix $A$ are linearly independent, then they span $\RR^n$.
	\item A square matrix with two identical columns is singular.
\end{enumerate}

\item {} [2.1.32] Let $A$ be an $(m\times n)$ matrix and let $D$ be an $(n\times m)$ matrix such that $AD=I_m$.  Show that for each $\vv b\in \RR^m$, the equation $A\vv x = \vv b$ has a solution.  [Hint: Think about the equation $AD\vv b = \vv b$.]  Also show that $m\le n$.

\item {} [2.2.\{39-42\}] Find the inverses of the following matrices, if they exist.  Use the row reduction algorithm.
\begin{multicols}{2}
\begin{enumerate}
	\item $\mtrix{2}{1 & 2 \\ 4 & 7}$
	\item $\mtrix{2}{5 & 10 \\ 4 & 7}$
	\item $\mtrix{3}{1 & 0 & -2 \\ -3 & 1 & 4\\ 2 & -3 & 4}$
	\item $\mtrix{3}{1 & -2 & 1 \\ 4 & -7 & 3 \\ -2 & 6 & -4}$
\end{enumerate}
\end{multicols}

\item {} [2.3.\{1,2,6,7,8\}] Using as few calculations as possible, determine if the following matrices are invertible.  (Do not fully compute any inverses.)  Justify your answers.
\begin{multicols}{2}
\begin{enumerate}
	\item $\mtrix{2}{5 & 7\\ -3 & -6}$
	\item $\mtrix{2}{-4 & 6\\ 6 & -9}$
	\item $\mtrix{3}{1 & -5 & -4\\ 0 & 3 & 4\\ -3 & 6 & 0}$
	\item $\mtrix{4}{-1 & -3 & 0 & 1\\ 3 & 5 & 8 & -3\\ -2 & -6 & 3 & 2 \\ 0 & -1 & 2 & 1}$
	\item $\mtrix{4}{1 & 3 & 7 & 8 \\ 0 & 2 & 1 & 5 \\ 0 & 0 & 8 & 8 \\ 0 & 0 & 0 & 3}$
\end{enumerate}
\end{multicols}


\end{enumerate}
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