Activity 2.2.5
Consider the matrices
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.
Compute the product \(AB\text{.}\)
Sage can multiply matrices using the
*
operator. Define the matrices \(A\) and \(B\) in the Sage cell below and check your work by computing \(AB\text{.}\)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 \(AB = BA\text{?}\)

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

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 \(B = C\text{?}\) 
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 \(B=0\text{?}\)