The table shows the prices of movie tickets categorised by different days of the week and the various age groups.

1. Write the ticket prices in matrix form using the given row and column categories.

Rows: Child, Adult, Senior Citizen.
Columns: Monday to Thursday, Friday, Saturday and Sunday.

$$P=\begin{pmatrix} 6.00&8.50&9.00\\ 7.00&10.00&12.50\\ 6.50&9.00&10.00 \end{pmatrix}$$

1. Form the $2\times3$ matrix using families as rows and age groups as columns.

Columns are in the order: Child, Adult, Senior Citizen.

Ng family: $1$ child, $2$ adults, $1$ senior citizen $\Rightarrow (1,2,1)$.

Tan family: $3$ children, $2$ adults, $0$ senior citizens $\Rightarrow (3,2,0)$.

$$Q=\begin{pmatrix} 1&2&1\\ 3&2&0 \end{pmatrix}$$

1. Multiply $Q$ and $P$ using row-by-column products.

Using

$$Q=\begin{pmatrix}1&2&1\\3&2&0\end{pmatrix},\quad P=\begin{pmatrix} 6.00&8.50&9.00\\ 7.00&10.00&12.50\\ 6.50&9.00&10.00 \end{pmatrix},$$ $$QP=\begin{pmatrix} 1(6.00)+2(7.00)+1(6.50)&1(8.50)+2(10.00)+1(9.00)&1(9.00)+2(12.50)+1(10.00)\\ 3(6.00)+2(7.00)+0(6.50)&3(8.50)+2(10.00)+0(9.00)&3(9.00)+2(12.50)+0(10.00) \end{pmatrix}$$ $$=\begin{pmatrix} 26.50&37.50&44.00\\ 32.00&45.50&52.00 \end{pmatrix}.$$

2. Explain the meaning of the elements of $QP$.

Since rows of $Q$ represent the families and columns of $P$ represent the day categories, the entries of $QP$ represent the total ticket costs for each family on each type of day.

Rows: Ng family, Tan family.

Columns: Monday to Thursday, Friday, Saturday and Sunday.