In grocery mart G, water costs $1.80 per litre, fresh milk costs$2.90 per litre and juice costs $2.30 per litre. In grocery mart H, water costs $0.20 more per litre, fresh milk costs$0.40 less per litre and juice costs $0.10 less per litre. This information can be represented by the matrix $Q =$ (see figure).

1. Arrange the purchase quantities into a $2\times3$ matrix using the item order from $Q$.

Use rows for the two shoppers, Rayyen then Zinnie, and columns for $W$, $M$, $J$.

Rayyen: $4$ litres of water, $2$ litres of milk, $3$ litres of juice $\Rightarrow (4,2,3)$.

Zinnie: $3$ litres of water, $0$ litres of milk, $4$ litres of juice $\Rightarrow (3,0,4)$.

$P=\begin{pmatrix}4&2&3\\3&0&4\end{pmatrix}$

1. Set up the product $PQ$ using the matrices $P$ and $Q$.

From part (a),
$P=\begin{pmatrix}4&2&3\\3&0&4\end{pmatrix},\quad Q=\begin{pmatrix}1.8&0.2\\2.9&-0.4\\2.3&-0.1\end{pmatrix}$

So
$R=PQ=\begin{pmatrix}4&2&3\\3&0&4\end{pmatrix}\begin{pmatrix}1.8&0.2\\2.9&-0.4\\2.3&-0.1\end{pmatrix}.$

2. Calculate each element of $R$ by row-by-column multiplication.

3. Write the resulting $2\times2$ matrix.

$R=\begin{pmatrix}19.9&-0.3\\14.6&0.2\end{pmatrix}.$

1. Interpret the second column of the product matrix $R$.

In $Q$, the second column gives the price difference per litre for mart H compared with mart G.

Since $R=PQ$, the second column of $R$ gives the total price difference for each person's purchases:
$\begin{pmatrix}-0.3\\0.2\end{pmatrix}$
So Rayyen's total is $0.30 less at H than at G, while Zinnie's total is $0.20 more at H than at G.

1. Find Rayyen's total cost before discount at grocery mart H using matrix R.

From the first row of $R$, Rayyen's total cost at mart G is $20.90 and the change in cost at mart H is $-\char36{}0.30$.

So his total cost at mart H before discount is
$20.90-0.30=20.60.$

2. Apply the 15% discount to Rayyen's mart H total.

A discount of $15\%$ means Rayyen pays $85\%$ of the price.

$0.85\times 20.60=17.51$