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.)

  • $\begingroup$ 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. $\endgroup$ – 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.


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