##### Activity2.2.3

In addition, Sage can find the product of a matrix and vector using the * operator. For example,

1. Use Sage to evaluate the product Item a yet again.

2. In Sage, define the matrix and vectors

\begin{equation*} A = \left[ \begin{array}{rrr} -2 \amp 0 \\ 3 \amp 1 \\ 4 \amp 2 \\ \end{array} \right], \zerovec = \left[ \begin{array}{r} 0 \\ 0 \end{array} \right], \vvec = \left[ \begin{array}{r} -2 \\ 3 \end{array} \right], \wvec = \left[ \begin{array}{r} 1 \\ 2 \end{array} \right] \text{.} \end{equation*}

3. What do you find when you evaluate $$A\zerovec\text{?}$$

4. What do you find when you evaluate $$A(3\vvec)$$ and $$3(A\vvec)$$ and compare your results?

5. What do you find when you evaluate $$A(\vvec+\wvec)$$ and $$A\vvec + A\wvec$$ and compare your results?

6. If $$I=\left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]$$ is the $$3\times3$$ identity matrix, what is the product $$IA\text{?}$$

in-context