I'm having trouble interpreting the results of matrix multiplication. For example,
Matrix $A$ gives the time that is required for two different $B$akeries to make a donut and a scone.
$A= \begin{array}{c c} & \begin{array}{c c } B_1 & B_2 \\ \end{array} \\ \begin{array}{c c }Donuts\\Scones\\ \end{array} & \left[\begin{array}{c c } 5 & 6 \\ 10 & 8 \\ \end{array} \right] \end{array}$So my row labels are donuts and scones and my column labels are $B_1$ and $B_2$. I'll call these snacks.
Matrix $B$ give the orders from two different $C$ustomers for donuts and scones. $B= \begin{array}{c c} & \begin{array}{c c } Donuts & Scones \\ \end{array} \\ \begin{array}{c c }Customer_1\\Customer_2\\ \end{array} & \left[\begin{array}{c c } 5 \quad & 6 \\ 10 \quad & 8 \\ \end{array} \right] \end{array}$
So my row labels are $Customer_1$ and $Customer_2$ and my column labels are Donuts and Scones.
If I multiply $A$ and $B$, I get
$R = \begin{bmatrix} 700 & 540 \\ 1100 & 920 \end{bmatrix}$
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:
$R = \begin{bmatrix} 850 & 780 \\ 775 & 770 \end{bmatrix}$
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 $B_1$ in $850$ minutes.
Am I correct?
Are the quantities in matrix multiplication $\boxed{M_1\text{-row-column-unit--}M_2\text{-column-row-unit}}$?
(In Given Example, the "units" are orders and time respectively.)
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.