A cake shop sells 3 types of cakes.
On Day 1, the shop sold 84 chocolate cakes, 90 vanilla cakes and 56 strawberry cakes.
On Day 2, the shop sold 92 chocolate cakes, 60 vanilla cakes and 61 strawberry cakes.

1. Place the given sales data into a $2\times 3$ matrix in the correct order.

Use rows for Day 1 and Day 2, and columns for Chocolate, Vanilla and Strawberry.

$P=\begin{pmatrix}84&90&56\\92&60&61\end{pmatrix}$

1. Multiply the $2\times3$ sales matrix by the $3\times2$ price-and-profit matrix.

Using $P=\begin{pmatrix}84&90&56\\92&60&61\end{pmatrix},$

$$R=\begin{pmatrix}84&90&56\\92&60&61\end{pmatrix} \begin{pmatrix}7&4\\8&3\\9&5\end{pmatrix}$$ $$=\begin{pmatrix} 84(7)+90(8)+56(9) & 84(4)+90(3)+56(5)\\ 92(7)+60(8)+61(9) & 92(4)+60(3)+61(5) \end{pmatrix}$$ $$=\begin{pmatrix} 1812&886\\ 1691&859 \end{pmatrix}.$$

1. Calculate the profit for Day 1 as a percentage of the total sale for Day 1.

From part (b), for Day 1,

Total sale $=1812$, profit $=886$.

$\frac{886}{1812}\times 100\%=48.896\ldots\%$

$\approx 48.9\%$

1. Substitute the matrix $R$ and carry out the row-by-matrix multiplication, then multiply by $\frac12$.

From part (b),
$R=\begin{pmatrix}1812&886\\1691&859\end{pmatrix}.$

$M=\frac12(1\;1)\begin{pmatrix}1812&886\\1691&859\end{pmatrix}$

$=\frac12\begin{pmatrix}1812+1691&886+859\end{pmatrix}$

$=\frac12\begin{pmatrix}3503&1745\end{pmatrix}$

$=\begin{pmatrix}1751.5&872.5\end{pmatrix}$

1. Interpret the two elements of $M$ from the columns of $R$.

In $R=\begin{pmatrix}1812&886\\1691&859\end{pmatrix}$, the first column represents total sales and the second column represents total profits for the two days.

Since
$M=\frac12(1\;1)R=\begin{pmatrix}1751.5&872.5\end{pmatrix},$
$M$ gives the average of the two days for each column.

So $1751.5$ is the average daily total sale, and $872.5$ is the average daily profit.