# Interpreting result of matrix multiplication

I'm having trouble interpreting the results of matrix multiplication. For example, matrix A gives the time that is required for two different bakeries to make a donut and a scone:

$$A = \begin{matrix} 5 & 6 \\ 10 & 8 \end{matrix}$$

So my row labels are donuts and scones and my column labels are B1 and B2. (I don't know how to add labels in mathjax.) . I'll call these snacks.

Matrix B give the orders from two different customers for donuts and scones:

$$B = \begin{matrix} 50 & 60 \\ 75 & 40 \end{matrix}$$

So my row labels are C1 and C2 and my column labels are donuts and scones.

If I multiply A and B I get

$$R = \begin{matrix} 700 & 540 \\ 1100 & 920 \end{matrix}$$

What do these quantities represent? Since I multiplied row of A against columns of B, I assume the resulting quantities represent $$A_{rows}B_{columns}$$ of "stuff". So it seems that I have 700 "snacks-bakery-time" (from A) of "snacks-customer-orders" (from B). This doesn't seem to make much sense, but matrix multiplication does not always make sense.

If instead, I had multiplied B times A, I would have gotten $$B_{rows}A_{columns}$$ of "stuff". That multiplication results in this matrix:

$$R = \begin{matrix} 850 & 780 \\ 775 & 770 \end{matrix}$$

So, in this case, 850 is "customer-snacks-order--bakery-snacks-time". Put another way, it means that customer 1's donut order can be produced by bakery 1 in 850 minutes.

Am I correct? Are the quantities in a matrix multiplication $$M_1$$-row-column-unit--$$M_2$$-column-row-unit? (Above the "units" are orders and time respectively.)

• Hint: For AB to make sense, if rows of A denote time, then column of B should denote rate (i.e., 1/t) of some quantity "q", so that their product denotes "q". If AB makes sense, then BA need not always make sense. – spkakkar Feb 6 '19 at 22:21

There's no reason for the numbers of customers, snacks and bakeries to be the same. You can multiply $$AB$$ if the number of columns of $$A$$ is the same as the number of rows of $$B$$.
Your matrix product $$AB$$ would make sense if the columns of $$A$$ and the rows of $$B$$ were labelled by the snacks, which seems to be the reverse of what you have. Then if the entry of $$A$$ in row $$i$$, column $$j$$ is the time for bakery $$i$$ to bake one item of snack $$j$$, and the entry of $$B$$ in row $$j$$, column $$k$$ is the number of snack $$j$$ ordered by customer $$k$$, their product is the time for bakery $$i$$ to bake the snack $$j$$'s ordered by customer $$k$$. Add that up over all snacks $$j$$ and you find that the entry of $$AB$$ in row $$i$$, column $$k$$ is the total time needed for bakery $$i$$ to fill customer $$k$$'s order.