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..

1. [1.9] True/False. Justify your answer.

   1. If $A$ is a $3\times 2$ matrix, then the transformation $x\mapsto \mathit{𝐴𝑥}$ cannot be one-to-one.
   2. If $A$ is a $3\times 2$ matrix, then the transformation $x\mapsto \mathit{𝐴𝑥}$ cannot map ${ℝ}^2$ onto ${ℝ}^3$.

2. [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.

   1. 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?
   2. Given that 70 million people live on the island, do there exist 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.

3. [2.1.1] Compute each matrix sum or product if it is defined. If undefined, then explain why.

   $$\begin{array}{llllllllllllllll}\hfill A& =\left[\begin{array}{ccc}\hfill 2& \hfill 0& \hfill -1\\ \hfill 4& \hfill -3& \hfill 2\end{array}\right]& \hfill B& =\left[\begin{array}{ccc}\hfill 7& \hfill -5& \hfill 1\\ \hfill 1& \hfill -4& \hfill -3\end{array}\right]& \hfill C& =\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill -2& \hfill 1\end{array}\right]& \hfill D& =\left[\begin{array}{cc}\hfill 3& \hfill 5\\ \hfill -1& \hfill 4\end{array}\right]& \hfill & & \hfill & & \hfill & & \hfill & \end{array}$$
   1. $-2A$
   2. $B-2A$
   3. $\mathit{𝐴𝐶}$
   4. $\mathit{𝐶𝐷}$
4. [2.1.11] Let $A=\left[\begin{array}{ccc}\hfill 1& \hfill 1& \hfill 1\\ \hfill 1& \hfill 2& \hfill 3\\ \hfill 1& \hfill 4& \hfill 5\end{array}\right]$ and $D=\left[\begin{array}{ccc}\hfill 2& \hfill 0& \hfill 0\\ \hfill 0& \hfill 3& \hfill 0\\ \hfill 0& \hfill 0& \hfill 5\end{array}\right]$. Compute $\mathit{𝐴𝐷}$ and $\mathit{𝐷𝐴}$. 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 $\mathit{𝐴𝐵}=\mathit{𝐵𝐴}$.
5. [2.1.9] Let $A=\left[\begin{array}{cc}\hfill 2& \hfill 5\\ \hfill -3& \hfill 1\end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 4& \hfill -5\\ \hfill 3& \hfill k\end{array}\right]$. What value(s) of $k$, if any, will make $\mathit{𝐴𝐵}=\mathit{𝐵𝐴}$?
6. [2.1.12] Let $A=\left[\begin{array}{cc}\hfill 3& \hfill -6\\ \hfill -1& \hfill 2\end{array}\right]$. Construct a $2\times 2$ matrix $B$ such that $\mathit{𝐴𝐵}$ is the zero matrix. Use two different nonzero columns for $B$.