A theatre sells tickets for a musical performance based on different categories. The table below shows the number of tickets sold for two consecutive nights for week 1.

1. Arrange the table entries in a $2\times 3$ matrix with the same row and column order.

Rows represent Saturday and Sunday respectively. Columns represent Cat 1, Cat 2 and Cat 3 respectively.

$M=\begin{pmatrix}430&635&335\\430&585&310\end{pmatrix}$

1. Arrange the three ticket prices in a $3\times 1$ matrix in category order.

Since the prices are for Cat 1, Cat 2 and Cat 3 respectively,
$P=\begin{pmatrix}98\\78\\48\end{pmatrix}$

1. Set up the matrix product $MP$ using the matrices from earlier parts.

$T=MP=\begin{pmatrix}430&635&335\\430&585&310\end{pmatrix}\begin{pmatrix}98\\78\\48\end{pmatrix}$

2. Calculate each element by multiplying each row of $M$ by the column of $P$.

$T=\begin{pmatrix}430(98)+635(78)+335(48)\\430(98)+585(78)+310(48)\end{pmatrix}$

$=\begin{pmatrix}42140+49530+16080\\42140+45630+14880\end{pmatrix}$

$=\begin{pmatrix}107750\\102650\end{pmatrix}$

1. Interpret each element of $T$ according to the row order of $M$.

Since
$T=\begin{pmatrix}107750\\102650\end{pmatrix},$
the first row corresponds to Saturday and the second row corresponds to Sunday.

Thus, $107750$ is the total amount collected from ticket sales on Saturday, and $102650$ is the total amount collected from ticket sales on Sunday.

1. Choose a matrix that adds the Saturday and Sunday rows of $M$ category by category.

To get the total for each category over both nights, add the two rows of $M$.

So $X$ must be the row matrix
$X=\begin{pmatrix}1&1\end{pmatrix}.$
Then
$XM=\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}430&635&335\\430&585&310\end{pmatrix},$
which adds the two rows of $M$.