##### Activity2.2.5

Consider the matrices

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 3 \amp 2 \\ -3 \amp 4 \amp -1 \\ \end{array}\right], B = \left[\begin{array}{rr} 3 \amp 0 \\ 1 \amp 2 \\ -2 \amp -1 \\ \end{array}\right] \text{.} \end{equation*}
1. Suppose we want to form the product $$AB\text{.}$$ Before computing, first explain how you know this product exists and then explain what the dimensions of the resulting matrix will be.

2. Compute the product $$AB\text{.}$$

3. Sage can multiply matrices using the * operator. Define the matrices $$A$$ and $$B$$ in the Sage cell below and check your work by computing $$AB\text{.}$$

4. Are you able to form the matrix product $$BA\text{?}$$ If so, use the Sage cell above to find $$BA\text{.}$$ Is it generally true that $$AB = BA\text{?}$$

5. Suppose we form the three matrices.

\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ 3 \amp -2 \\ \end{array}\right], B = \left[\begin{array}{rr} 0 \amp 4 \\ 2 \amp -1 \\ \end{array}\right], C = \left[\begin{array}{rr} -1 \amp 3 \\ 4 \amp 3 \\ \end{array}\right] \text{.} \end{equation*}

Compare what happens when you compute $$A(B+C)$$ and $$AB + AC\text{.}$$ State your finding as a general principle.

6. Compare the results of evaluating $$A(BC)$$ and $$(AB)C$$ and state your finding as a general principle.

7. When we are dealing with real numbers, we know if $$a\neq 0$$ and $$ab = ac\text{,}$$ then $$b=c\text{.}$$ Define matrices

\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ -2 \amp -4 \\ \end{array}\right], B = \left[\begin{array}{rr} 3 \amp 0 \\ 1 \amp 3 \\ \end{array}\right], C = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 2 \\ \end{array}\right] \end{equation*}

and compute $$AB$$ and $$AC\text{.}$$

If $$AB = AC\text{,}$$ is it necessarily true that $$B = C\text{?}$$

8. Again, with real numbers, we know that if $$ab = 0\text{,}$$ then either $$a = 0$$ or $$b=0\text{.}$$ Define

\begin{equation*} A = \left[\begin{array}{rr} 1 \amp 2 \\ -2 \amp -4 \\ \end{array}\right], B = \left[\begin{array}{rr} 2 \amp -4 \\ -1 \amp 2 \\ \end{array}\right] \end{equation*}

and compute $$AB\text{.}$$

If $$AB = 0\text{,}$$ is it necessarily true that either $$A=0$$ or $$B=0\text{?}$$

in-context