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