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. True/False. In the following, $A$, $B$, and $C$ are matrices for which the given expressions are defined. Justify your answer.

   1. Each column of $\mathit{𝐴𝐵}$ is a linear combination of the columns of $B$ using weights from the corresponding column of $A$.
   2. The second row of $\mathit{𝐴𝐵}$ is the second row of $A$ multiplied on the right by $B$.
   3. $A^T+B^T={(A+B)}^T$.
   4. $(\mathit{𝐴𝐵})C=(\mathit{𝐴𝐶})B$
   5. ${(\mathit{𝐴𝐵})}^T=A^T{B}^T$
   6. If $A$ can be row reduced to the identity matrix, then $A$ must be invertible.
   7. Each elementary matrix is invertible.
   8. If $A$ is invertible, then the elementary row operations that reduce $A$ to the identity $I_n$ also reduce $A^{-1}$ to $I_n$.
   9. If the columns of an $n\times n$ matrix $A$ are linearly independent, then they span ${ℝ}^n$.
   10. A square matrix with two identical columns is singular.
2. [2.1.32] Let $A$ be an $(m\times n)$ matrix and let $D$ be an $(n\times m)$ matrix such that $\mathit{𝐴𝐷}=I_m$. Show that for each $𝐛\in {ℝ}^m$, the equation $A𝐱=𝐛$ has a solution. [Hint: Think about the equation $\mathit{𝐴𝐷}𝐛=𝐛$.] Also show that $m\le n$.

3. [2.2.{39-42}] Find the inverses of the following matrices, if they exist. Use the row reduction algorithm.

   1. $\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 4& \hfill 7\end{array}\right]$
   2. $\left[\begin{array}{cc}\hfill 5& \hfill 10\\ \hfill 4& \hfill 7\end{array}\right]$
   3. $\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill -2\\ \hfill -3& \hfill 1& \hfill 4\\ \hfill 2& \hfill -3& \hfill 4\end{array}\right]$
   4. $\left[\begin{array}{ccc}\hfill 1& \hfill -2& \hfill 1\\ \hfill 4& \hfill -7& \hfill 3\\ \hfill -2& \hfill 6& \hfill -4\end{array}\right]$

4. [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.

   1. $\left[\begin{array}{cc}\hfill 5& \hfill 7\\ \hfill -3& \hfill -6\end{array}\right]$
   2. $\left[\begin{array}{cc}\hfill -4& \hfill 6\\ \hfill 6& \hfill -9\end{array}\right]$
   3. $\left[\begin{array}{ccc}\hfill 1& \hfill -5& \hfill -4\\ \hfill 0& \hfill 3& \hfill 4\\ \hfill -3& \hfill 6& \hfill 0\end{array}\right]$
   4. $\left[\begin{array}{cccc}\hfill -1& \hfill -3& \hfill 0& \hfill 1\\ \hfill 3& \hfill 5& \hfill 8& \hfill -3\\ \hfill -2& \hfill -6& \hfill 3& \hfill 2\\ \hfill 0& \hfill -1& \hfill 2& \hfill 1\end{array}\right]$
   5. $\left[\begin{array}{cccc}\hfill 1& \hfill 3& \hfill 7& \hfill 8\\ \hfill 0& \hfill 2& \hfill 1& \hfill 5\\ \hfill 0& \hfill 0& \hfill 8& \hfill 8\\ \hfill 0& \hfill 0& \hfill 0& \hfill 3\end{array}\right]$