两家药房销售免洗洗手液、洗手液和湿纸巾。矩阵 P 表示这两家药房可供销售的商品数量。 免洗洗手液 洗手液 湿纸巾 P = \begin{pmatrix}100 & 80 & 150 \ 60 & 75 & 120\end{pmatrix} （第一行：药房 A；第二行：药房 B） 为这两家药房提供产品的同一供应商对免洗洗手液的定价为每瓶 $3.50，对洗手液的定价为每瓶 $3，对湿纸巾的定价为每包 $1.50。 两家药房的所有免洗洗手液、洗手液和湿纸巾均已售罄。药房 A 获得了 20% 的利润，药房 B 获得了 25% 的利润。 将此信息表示为一个 3 x 1 的列矩阵 Q。 答案 Q = ( ) 计算矩阵 R = PQ。 答案 R = 说明矩阵 R 中的元素代表什么。 ... 计算矩阵 S = $\displaystyle\frac{1}{100}\begin{pmatrix}20 & 25\end{pmatrix} R$。 答案 S = 说明矩阵 S 代表什么。 ...

1. Form the $3\times1$ column matrix of supplier prices in the correct item order.

The columns of $P$ are in the order: Hand Sanitisers, Hand Wash, Wipes.

So the corresponding prices are $\char36{}3.50$, $\char36{}3.00$, $\char36{}1.50$.

$Q=\begin{pmatrix}3.50\\3.00\\1.50\end{pmatrix}$

1. Compute each element of $R=PQ$ using row-by-column multiplication.

$R=\begin{pmatrix}100&80&150\\60&75&120\end{pmatrix}\begin{pmatrix}3.50\\3.00\\1.50\end{pmatrix}$

$=\begin{pmatrix}100(3.50)+80(3.00)+150(1.50)\\60(3.50)+75(3.00)+120(1.50)\end{pmatrix}$

1. Interpret each element of $R$ according to the rows of matrix $P$ and the price column matrix $Q$.

Since $P$ gives the number of items in Pharmacy A and Pharmacy B, and $Q$ gives the supplier price per item, the product
$R=PQ=\begin{pmatrix}815\\615\end{pmatrix}$
gives the total supplier cost for each pharmacy.

So the first element, $815$, is the total cost price for Pharmacy A, and the second element, $615$, is the total cost price for Pharmacy B.

1. Substitute $R$ and evaluate the row-by-column product.

Using $R=\begin{pmatrix}815\\615\end{pmatrix}$,
$S=\frac{1}{100}\begin{pmatrix}20&25\end{pmatrix}\begin{pmatrix}815\\615\end{pmatrix}$
$=\frac{1}{100}\begin{pmatrix}20(815)+25(615)\end{pmatrix}$

2. Complete the arithmetic to find matrix $S$.

$S=\frac{1}{100}\begin{pmatrix}16300+15375\end{pmatrix}$
$=\frac{1}{100}\begin{pmatrix}31675\end{pmatrix}$
$=\begin{pmatrix}316.75\end{pmatrix}$

1. Interpret the value of $S$ in the context of the profit percentages and total costs.

Since $S=\frac{1}{100}\begin{pmatrix}20&25\end{pmatrix}\begin{pmatrix}815\\615\end{pmatrix},$ it calculates $20\%\text{ of Pharmacy A's cost}+25\%\text{ of Pharmacy B's cost}.$ Hence $S$ represents the total profit made by both pharmacies.