VCE General Maths Matrices 2023 Exam 2 Mini Test

Question 1 (3 marks) [2023 Exam 2 Q8]

A circus sells three different types of tickets: family (\(F\)), adult (\(A\)) and child (\(C\)).

The cost of admission, in dollars, for each ticket type is presented in matrix \(N\) below.

\[ N = \begin{bmatrix} 36 \\ 15 \\ 8 \end{bmatrix} \begin{matrix} F \\ A \\ C \end{matrix} \]

The element in row \(i\) and column \(j\) of matrix \(N\) is \(n_{ij}\).

a. Which element shows the cost for one child ticket? 1 mark

b. A family ticket will allow admission for two adults and two children.
Complete the matrix equation below to show that purchasing a family ticket could give families a saving of $10. 1 mark

[ 0    2    2 ] × \(N\) − [ ] × \(N\) = [ 10 ]

c. On the opening night, the circus sold 204 family tickets, 162 adult tickets and 176 child tickets.
The owners of the circus want a 3 × 1 product matrix that displays the revenue for each ticket type: family, adult and child.
This product matrix can be achieved by completing the following matrix multiplication.

\[ K \times N = \begin{bmatrix} 7344 \\ 2430 \\ 1408 \end{bmatrix} \]

Write down matrix \(K\) in the space below. 1 mark

\(K = \)

Question 2 (4 marks) [2023 Exam 2 Q9]

The circus is held at five different locations, \(E, F, G, H\) and \(I\).

The table below shows the total revenue for the ticket sales, rounded to the nearest hundred dollars, for the last 20 performances held at each of the five locations.

Location	E	F	G	H	I
Ticket sales	$960 000	$990 500	$940 100	$920 800	$901 300

The ticket sales information is presented in matrix \(R\) below.

\(R = [960\,000 \quad 990\,500 \quad 940\,100 \quad 920\,800 \quad 901\,300]\)

a. Complete the matrix equation below that calculates the average ticket sales per performance at each of the five locations. 1 mark

[ ] × \(R\) = [ ]

The circus would like to increase its total revenue from the ticket sales from all five locations.

The circus will use the following matrix calculation to target the next 20 performances.

\[ [t] \times R \times \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \]

b. Determine the value of \(t\) if the circus would like to increase its revenue from ticket sales by 25%. 1 mark

c. The circus moves from one location to the next each month. It rotates through each of the five locations, before starting the cycle again.

The following matrix displays the movement between the five locations.

\[ \begin{array}{c@{}c} & \begin{array}{c} \textit{this month} \\ \begin{array}{@{}ccccc@{}} E & F & G & H & I \end{array} \end{array} \\ \begin{matrix} E \\ F \\ G \\ H \\ I \end{matrix} & \left[ \begin{array}{@{}ccccc@{}} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \end{array} \right] \end{array} \quad \textit{next month} \]

The circus plans to add a sixth location, \(J\).

The only change to the cycle is that the circus will be held at location \(J\) after location \(E\) and before location \(G\).

d. Complete the three columns in the following matrix, showing the new movement between the six locations, \(E, F, G, H, I\) and \(J\). 1 mark

\[ \begin{array}{c@{}c} & \begin{array}{c} \textit{this month} \\ \begin{array}{@{}cccccc@{}} E & F & G & H & I & J \end{array} \end{array} \\ \begin{matrix} E \\ F \\ G \\ H \\ I \\ J \end{matrix} & \left[ \begin{array}{@{}cccccc@{}} \underline{\hspace{0.5cm}} & 0 & \underline{\hspace{0.5cm}} & 1 & 0 & \underline{\hspace{0.5cm}} \\ \underline{\hspace{0.5cm}} & 0 & \underline{\hspace{0.5cm}} & 0 & 0 & \underline{\hspace{0.5cm}} \\ \underline{\hspace{0.5cm}} & 0 & \underline{\hspace{0.5cm}} & 0 & 0 & \underline{\hspace{0.5cm}} \\ \underline{\hspace{0.5cm}} & 0 & \underline{\hspace{0.5cm}} & 0 & 1 & \underline{\hspace{0.5cm}} \\ \underline{\hspace{0.5cm}} & 1 & \underline{\hspace{0.5cm}} & 0 & 0 & \underline{\hspace{0.5cm}} \\ \underline{\hspace{0.5cm}} & 0 & \underline{\hspace{0.5cm}} & 0 & 0 & \underline{\hspace{0.5cm}} \end{array} \right] \end{array} \quad \textit{next month} \]

Question 3 (3 marks) [2023 Exam 2 Q10]

Within the circus, there are different types of employees: directors (\(D\)), managers (\(M\)), performers (\(P\)) and sales staff (\(S\)). Customers (\(C\)) attend the circus.

Communication between the five groups depends on whether they are customers or employees, and on what type of employee they are.

Matrix \(G\) below shows the communication links between the five groups.

\[ \begin{array}{cc} & \begin{array}{c} \mathrm{receiver} \\ \begin{array}{ccccc} D & M & P & S & C \end{array} \end{array} \\ G = \mathrm{sender} \begin{matrix} D \\ M \\ P \\ S \\ C \end{matrix} & \left[ \begin{array}{ccccc} 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right] \end{array} \]

In this matrix:

• • The ‘1’ in row \(D\), column \(M\) indicates that the directors can communicate directly with the managers.
• • The ‘0’ in row \(P\), column \(D\) indicates that the performers cannot communicate directly with the directors.

a. A customer wants to make a complaint to a director.
What is the shortest communication sequence that will successfully get this complaint to a director? 1 mark

b. Matrix \(H\) below shows the number of two-step communication links between each group. Sixteen elements in this matrix are missing.

\[ \begin{array}{cc} & \begin{array}{c} \mathrm{receiver} \\ \begin{array}{ccccc} D & M & P & S & C \end{array} \end{array} \\ H = \mathrm{sender} \begin{matrix} D \\ M \\ P \\ S \\ C \end{matrix} & \left[ \begin{array}{ccccc} 1 & \_ & \_ & \_ & \_ \\ 0 & \_ & \_ & \_ & \_ \\ 1 & \_ & \_ & \_ & \_ \\ 1 & \_ & \_ & \_ & \_ \\ 0 & 1 & 0 & 0 & 1 \end{array} \right] \end{array} \]

i. Complete matrix \(H\) above by filling in the missing elements. 1 mark

ii. What information do elements \(g_{21}\) and \(h_{21}\) provide about the communication between the circus employees? 1 mark