Print this problem set out (there are X problems!) and answer the problems on the given sheet.

1. Consider the matrix $$ A_1 = \begin{bmatrix} -7 & 3 & 0\\ 2 & -3 & 8\\ 5 & 5 & 1\\ -1 & 0 & 4\\ \end{bmatrix} $$
   1. What is the dimension of this matrix?
   2. What are the row vectors of this matrix?
   3. What are the column vectors of his matrix?
   4. In the equation $A_1\cdot\boldsymbol{x}=\boldsymbol{0}$, what is the dimension of vector $\boldsymbol{x}$?
   5. Write down the system of linear equations $A_1\cdot\boldsymbol{x}=\boldsymbol{0}$ represents:
2. What is the result of the following matrix-vector product: $$ \begin{bmatrix} -7 & 3 & 0 & 5\\ 2 & -3 & 8 & -1\\ 5 & 5 & 1 & -7\\ \end{bmatrix} \cdot \begin{bmatrix} -2\\ -4\\ 1\\ 3 \end{bmatrix} = $$
3. Completely describe the solutions to each of the equations below. Note that the matrices are already in row echelon form.
   1. Equation 1: $$ \begin{bmatrix} 2 & -1 & 3\\ 0 & 3 & 12\\ 0 & 0 & 4 \end{bmatrix} \cdot \boldsymbol{x} = \boldsymbol{0} $$
   2. Equation 2: $$ \begin{bmatrix} 5 & -1 & 3 & 4\\ 0 & 2 & -1 & -9\\ 0 & 0 & 0 & 3 \end{bmatrix} \cdot \boldsymbol{x} = \boldsymbol{0} $$


4. Solve the following matrix-vector product equation: $$ \begin{bmatrix} 2 &-1 & 3 \\ 4 &-8 & 9 \\ -1 &-17.5 & 10.5 \end{bmatrix} \boldsymbol{x} = \begin{bmatrix} 2\\ 4\\ -7 \end{bmatrix} $$
5. Compute the following matrix product: $$ \begin{bmatrix} -2 & 0 & 3\\ 1 & 5 & -1\\ 0 & 4 & 1\\ 9 & -1 & 3 \end{bmatrix} \cdot \begin{bmatrix} 3 & 5\\ 2 & -1\\ 1 & 1 \end{bmatrix} = $$
6. Compute the following matrix product: $$ \begin{bmatrix} a & b & c \\ d & e & f \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} u & v & w \\ x & y & z \\ 0 & 0 & 1 \end{bmatrix} = $$
7. What dimension would the result of the following product have: $ \underbrace{A}_{5\times 3} \cdot \underbrace{B}_{3\times 6} \cdot \underbrace{C}_{6\times 2} \cdot \underbrace{D}_{2\times 8} $ dim: _____________
8. What dimension would the result of the following product have: $ \underbrace{A}_{7\times 3} \cdot \underbrace{B}_{3\times 6} \cdot \underbrace{C}_{5\times 6} \cdot \underbrace{D}_{6\times 4} $ dim: _____________


9. The ring $\mathbb{Z}_3$ is defined as follows (you know this already, of course):

   The $\mathbb{Z}_3$ number system	"numbers" $\{0,1,2\}$	$+$ 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1	$*$ 0 1 2 0 0 0 0 1 0 1 2 2 0 2 1

   Do gaussian elimination to put the following matrix in row echelon form. Show your work!
   Note: this is over $\mathbb{Z}_3$! If you use fractions or decimals you are doing it wrong! The whole point here is to think about gaussian elimination carefully. Looking closely at the algorithm in the notes may help. $$ \begin{bmatrix} 1 & 0 & 1 & 1 \\ 2 & 1 & 1 & 1\\ 1 & 2 & 0 & 1 \end{bmatrix} $$