Hand sanitisers, hand wash and wipes were sold in two pharmacies. The matrix P shows the number of items available for sale in the two pharmacies. Hand Sanitisers Hand Wash Wipes P = \begin{pmatrix}100 & 80 & 150 \ 60 & 75 & 120\end{pmatrix} (First row: Pharmacy A; Second row: Pharmacy B) The same supplier producing the products for the two pharmacies charges the hand sanitisers at $3.50 per bottle, the hand wash at $3 per bottle and the wipes at $1.50 per pack. All the hand sanitisers, hand wash and wipes were sold out in both pharmacies. Pharmacy A made a profit of 20% and Pharmacy B made a profit of 25%. Represent this information in a 3 x 1 column matrix Q. Answer Q = ( ) Evaluate the matrix R = PQ. Answer R = State what the elements of matrix R represent. ... Evaluate the matrix S = $\displaystyle\frac{1}{100}\begin{pmatrix}20 & 25\end{pmatrix} R$. Answer S = State what matrix S represent. ...

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.