Q.A trader sells pens and notebooks from two shops. The quantity sold (in units) is given by $Q=\begin{pmatrix}100&50\\80&70\end{pmatrix}$, where the rows represent Shop A and Shop B and the columns represent Pens and Notebooks. If a pen costs Rs.,5 and a notebook costs Rs.,10, represented by $P=\begin{pmatrix}5\\10\end{pmatrix}$, use matrix multiplication to find the total revenue of each shop.

Total revenue per shop is obtained by multiplying the quantity matrix by the price column, since $Q$ is $2\times2$ and $P$ is $2\times1$, giving a conformable $2\times1$ product. …

$Q$ has order $2\times2$ and $P$ has order $2\times1$; since the number of columns of $Q$ ($=2$) equals the number of rows of $P$ ($=2$), the product $QP$ is defined and has order $2\times1$.

So Shop A's total revenue is Rs.,1000 and Shop B's is Rs.,1100 — each entry of $QP$ combines that shop's pen sales and notebook sales into one rupee figure, exactly the kind of consolidation matrix multiplication is used for in business. …

Trying to multiply $P$ by $Q$ as $PQ$ instead of $QP$ — since $P$ is $2\times1$ and $Q$ is $2\times2$, $PQ$ is not conformable ($1 \ne 2$) and is simply undefined; the order of multiplication must m …