The matrix A below shows the prices of football match tickets for seats in Category 1 (Cat 1), Category 2 (Cat 2) and Category 3 (Cat 3). The prices are given in dollars.

1. Represent the seat numbers as a 3×1 column matrix in category order

Cat 1, Cat 2, Cat 3 correspond to the three columns of matrix $A$, so
$B=\begin{pmatrix}\text{Cat 1 seats}\\\text{Cat 2 seats}\\\text{Cat 3 seats}\end{pmatrix}=\begin{pmatrix}300\\500\\1000\end{pmatrix}.$

1. Form and evaluate the product $AB$ using the ticket prices and seat numbers.

Using $B=\begin{pmatrix}300\\500\\1000\end{pmatrix},$
$AB=\begin{pmatrix}80&42&20\\120&62&30\end{pmatrix}\begin{pmatrix}300\\500\\1000\end{pmatrix}$
$=\begin{pmatrix}80(300)+42(500)+20(1000)\\120(300)+62(500)+30(1000)\end{pmatrix}$
$=\begin{pmatrix}65000\\97000\end{pmatrix}.$

2. Multiply the row matrix $(0.5\quad 0.5)$ by $AB$ to find $X$.

$X=\begin{pmatrix}0.5&0.5\end{pmatrix}\begin{pmatrix}65000\\97000\end{pmatrix}$
$=0.5(65000)+0.5(97000)$
$=81000.$

1. Interpret the element of $X$ in context.

From part (b), $X=(0.5\quad 0.5)AB=81000$. This is half the semi-final total takings plus half the final total takings, so it represents the average total ticket sales revenue per match.