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The question is:

Draw the cuts in A (2 by 3) and B (3 by 4) and AB to show how each of the four multiplication rules is really a block multiplication.

1.) Matrix A times the columns of B: Columns of AB
2.) Rows of A times the Matrix B: Rows of AB
3.) Rows of A times the columns of B: Inner products (numbers in AB)
4.) Columns of A times rows of B: Outer products (matrices added to AB)

Part of the problem is that I'm not 100% sure what is being asked here. If someone could explain how to solve 1 and 2 below and the differences between them then I think I'd be able to do the rest. I include my attempts at 1 and 2 below:

Q1.) This would be done 4 times: (2,3)*(3,1) = (2,1) * 4

$ \left[ \begin{array}{ccc} & & \\ & & \end{array} \right] \left[ \begin{array} \\ & \\ & \end{array} \right] = \left[ \begin{array}{c|c|c|c} & & & \\ & & & \end{array} \right] $

Q2.) This would be done 2 times: (1,3) * (3,4) = (1*4) *2

$ \left[ \begin{array}{ccc} & & \\ \end{array} \right] \left[ \begin{array}{cccc} & & & \\ & & &\\ & & & \end{array} \right] = \left[ \begin{array} & & & \\ \hline & & & \end{array} \right] $

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I believe Strang is just trying to illustrate how to visually perceive, at a glance, the rules of multiplication. I think he just wants you to consider the "blocks" determined by the stated partitioning in each question (i.e. row or columns)

We see that question 1 is simply asking to show $A$ times the "cuts" of $B$, where the blocks defining the cuts are the columns of $B$. So, we have:

$A \bigg[ \quad \bigg| \quad \bigg| \quad \bigg| \quad \bigg]$.

Question 2 just fixes matrix $B$. So:

$\bigg[ \begin{array} & & & \\ \hline & & & \end{array}\bigg]B$

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