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\headers{Math343}{Homework 7}{Due Oct 8 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 {} [1.9] True/False.  Justify your answer.
\begin{enumerate}
	\item If $A$ is a $3\times 2$ matrix, then the transformation $x\mapsto Ax$ cannot be one-to-one.
	\item If $A$ is a $3\times 2$ matrix, then the transformation $x\mapsto Ax$ cannot map $\RR^2$ onto $\RR^3$.
\end{enumerate}

\item {} [1.10] Two nations, $A$ and $B$, occupy an island.  Each year, 10\% of $A$'s population moves to $B$ and 25\% of $B$'s population moves to $A$.  The rest stay put.
\begin{enumerate}
	\item If $A$ begins with 30 million people and $B$ begins with 40 million people, what will their populations be after one, two, and three years?
	\item Given that 70 million people live on the island, do there exist \emph{stable} population levels for $A$ and $B$ that would stay the same year after year?  Either find stable population levels or explain why they do not exist.
\end{enumerate}

\item {} [2.1.1] Compute each matrix sum or product if it is defined.  If undefined, then explain why.
\begin{align*}
A &= \mtrix{3}{2 & 0 & -1\\ 4 & -3 & 2} &
B &= \mtrix{3}{7 & -5 & 1\\ 1 & -4 & -3} &
C &= \mtrix{2}{1 & 2 \\ -2 & 1} &
D &= \mtrix{2}{3 & 5 \\ -1 & 4}
\end{align*}
\begin{multicols}{2}
\begin{enumerate}
	\item $-2A$
	\item $B-2A$
	\item $AC$
	\item $CD$
\end{enumerate}
\end{multicols}

\item {} [2.1.11] Let $A=\mtrix{3}{1 & 1 & 1\\ 1 & 2 & 3 \\ 1 & 4 & 5}$ and $D=\mtrix{3}{2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 5}$.  Compute $AD$ and $DA$.  Explain how the columns or rows of $A$ change when $A$ is multiplied by $D$ on the right or on the left.  Find a $3\times 3$ matrix $B$, not the identity matrix or the zero matrix, such that $AB=BA$.

\item {} [2.1.9] Let $A=\mtrix{2}{2 & 5\\-3 & 1}$ and $B=\mtrix{2}{4 & -5 \\ 3 & k}$.  What value(s) of $k$, if any, will make $AB=BA$?

\item {} [2.1.12] Let $A=\mtrix{2}{3 & -6 \\ -1 & 2}$.  Construct a $2\times 2$ matrix $B$ such that $AB$ is the zero matrix.  Use two different nonzero columns for $B$.


\end{enumerate}
\end{document}
