Now compute \(D - A\text{.}\) Do this numerically, not algebraically; that is, forget where your result matrix \(D\) came from and actually compute \(D - A\) using the same procedure that you used to subtract matrices in Task a. What are the dimensions of this result?
Now let’s remember that \(D = A + C\text{.}\) Algebraically, what result would you expect from computing \((A + C) - A\text{?}\) Does your numerical computation in the previous step agree with your algebraic expectation? (Keep in mind your answer to what it means for two matrices to be equal from Task b of Discovery 4.2.)
Use your idea from Task b of Discovery 4.2 to turn the following single matrix equation into a system of two equations in the unknowns \(c\) and \(d\text{.}\) (Don’t bother to actually solve for the values of \(c\) and \(d\text{.}\))
\begin{equation*}
\begin{bmatrix} c + 2 d \\ 3 d \end{bmatrix}
= \begin{abmatrix}{r} 5 \\ -3 \end{abmatrix}
\end{equation*}
Again, be careful about the sizes of your matrices! If you have an equals sign between two matrices, they must adhere to your principle from Task b of Discovery 4.2.
On the left-hand side of the matrix equation \(A \uvec{x} = \uvec{b}\text{,}\) the operation matrix-times-matrix should compute to a single matrix. What size of matrix should this multiplication result be?
The result of computing \(A \uvec{x}\) must make sense in the matrix equality \(A \uvec{x} = \uvec{b}\text{,}\) per the pattern of matrix equality you described in Task b of Discovery 4.2.
Finally, we want \(A \uvec{x} = \uvec{b}\) to represent in one matrix equation the full system of two number equations from Task b. We already came up with a matrix equation to represent that system in Task b. Looking at your matrices \(A\) and \(\uvec{x}\) from Task c, and comparing with the left-hand side of your matrix equation from Task b, what procedure should be used to carry out the operation matrix \(A\) times column \(\uvec{x}\)?
The values \(x_1 = 2\text{,}\)\(x_2 = 1\text{,}\)\(x_3 = 3\text{,}\) represent a solution to the system in Task b. Verify this by carrying out the multiplication \(A \uvec{x}\text{,}\) using your calculation procedure from Task e, and with the unknowns \(x_1,x_2,x_3\) in the column matrix \(\uvec{x}\) replaced by these solution values. Then compare your calculation result with \(\uvec{b}\text{.}\)
We all know that \(3\) times \(5\) and \(5\) times \(3\) have the same result. Algebraically, we write that \(a b = b a \) is true for all numbers \(a,b\text{.}\) What about matrices?
Look back at matrices \(A\) and \(X\) from Discovery 4.6, where you computed the matrix product \(A X\text{.}\) Does multiplying \(X A\) in the opposite order even make sense?
Considering the previous three activities about matrix multiplication, what patterns have you observed about the required sizes of the two matrices involved for things to work out?
In particular, if \(A\) has \(m\) rows and \(n\) columns, and \(B\) has \(k\) rows and \(\ell\) columns, what relationship must there be between these numbers for the matrix-times-columns calculation method to make sense when computing \(A B\text{?}\) And in that case, what size will the resulting product matrix \(A B\) be?
