A concert offers three types of seating areas, A, B and C, for Saturday and Sunday. The numbers of tickets sold for each type of seating area on each of these days in a particular week are shown in the following table.

1. Write the ticket sales table as a $2\times3$ matrix, preserving the row and column order.

Rows correspond to Saturday and Sunday. Columns correspond to seating areas A, B and C.

$S=\begin{pmatrix}1100&1000&1200\\1400&1200&1300\end{pmatrix}$

1. Write the three ticket prices as a $3\times1$ column matrix in the order A, B, C.

Since the prices for seating areas A, B and C are $110, $80 and $x respectively,
$P=\begin{pmatrix}110\\80\\x\end{pmatrix}$

1. Set up the matrix product $SP$ using the matrices from parts (a) and (b).

$T=SP=\begin{pmatrix}1100&1000&1200\\1400&1200&1300\end{pmatrix}\begin{pmatrix}110\\80\\x\end{pmatrix}$

2. Carry out row-by-column multiplication and simplify each element in terms of $x$.

$T=\begin{pmatrix}1100(110)+1000(80)+1200x\\1400(110)+1200(80)+1300x\end{pmatrix}$

$=\begin{pmatrix}121000+80000+1200x\\154000+96000+1300x\end{pmatrix}$

$=\begin{pmatrix}201000+1200x\\250000+1300x\end{pmatrix}$

1. Interpret the two elements of $T$ using the row order from matrix $S$.

Since $T=SP$, each element is calculated by multiplying the number of tickets sold for A, B and C by their ticket prices and adding.

The first element corresponds to the Saturday row of $S$, and the second element corresponds to the Sunday row of $S$.

1. Form an equation using the total weekend collection.

From part (c),
$T=\begin{pmatrix}201000+1200x\\250000+1300x\end{pmatrix}$
The total collected for the weekend is
$(201000+1200x)+(250000+1300x)=688500.$

2. Solve the resulting linear equation for $x$.

$451000+2500x=688500$
$2500x=688500-451000=237500$
$x=\frac{237500}{2500}=95$