Matrix products in context

A quantity matrix multiplied by a price column gives totals. Match the inner categories in the same order; the output rows keep the quantity matrix’s row meanings.

Example

For East and West shops, \(S=\begin{pmatrix}12&5\\8&9\end{pmatrix}\) lists pens then notebooks. Prices are \(\$2\) per pen and \(\$4\) per notebook, so \(P=\begin{pmatrix}2\\4\end{pmatrix}\).

\(\begin{aligned}SP&=\begin{pmatrix}12(2)+5(4)\\8(2)+9(4)\end{pmatrix}\\&=\begin{pmatrix}44\\52\end{pmatrix}.\end{aligned}\)

East’s revenue is \(\$44\), West’s is \(\$52\). The product has one row per shop and one revenue column. Units are items multiplied by dollars per item, giving dollars.

If the notebook price is \(x\) dollars and East’s revenue is \(\$49\), its row gives \(24+5x=49\), so \(x=5\). Check: \(12(2)+5(5)=49\). A column of ones instead adds the quantities: \(S\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}17\\17\end{pmatrix}\).