VCAA General Mathematics Matrices

$F$ is a $4 \times 4$ matrix.

The element in row $i$ and column $j$ of $F$ is $f_{ij}$.

The elements are determined by the rule $f_{ij} = i^2 - j$.

How many of the elements in $F$ will be negative?

A species of bird has a life span of three years.

The females in this species do not reproduce in their first year but produce an average of four female offspring in their second year, and three in their third year.

The Leslie matrix, $L$, below is used to model the female population distribution of this species of bird.

The element in the second row, first column states that on average 20% of this population will

How many of the following statements are true?

• All square matrices have an inverse.
• The inverse of a matrix could be the same as the transpose of that matrix.
• If the determinant of a matrix is equal to zero, then the inverse does not exist.
• It is possible to take the inverse of an identity matrix.

Interpret the meaning of element $t_{14}$.

In the long term, what is predicted to happen to the number of resorts classified as Budget?

Let $S_0$ be the state matrix that shows the number of resorts classified as either $L$, $M$, $S$ or $B$ at the start of 2024.

Let $S_1$ be the state matrix that shows the expected number of resorts classified as either $L$, $M$, $S$ or $B$ at the start of 2025.

Using this information, determine the value of $d$ in transition matrix $T$.

What is the expected percentage growth in the number of Luxury resorts from the start of 2024 to the start of 2026?

Write your answer correct to one decimal place.

Consider the following matrix, where $h \neq 0$.

$\begin{bmatrix} 4 & g \\ 8 & h \end{bmatrix}$

The inverse of this matrix does not exist when $g$ is equal to

A course requires students to pass two levels of study before they graduate.

Each level of study takes one year.

• Students must pass the first level ($L1$) before they move on to the second level ($L2$).
• Students must pass the second level before they become graduates ($G$).

From one year to the next, it is known that

• 80% of students pass the first level of study and progress to the second level of study
• 60% of students in the second level of study become graduates the following year.

Students who do not pass a level of study repeat it in the following year. No student leaves the course before graduating.

What percentage of students are expected to graduate without repeating a level of study?

A market stall sells three types of candles.

The cost of each type of candle is shown in matrix $C$ below.

$C = \begin{bmatrix} 25 & 32 & 43 \end{bmatrix}$

Towards the end of the day, the cost of each item is discounted by 15%.

Which one of the following expressions can be used to determine each discounted price?

After one day, how many chickens are expected to remain in the same field that they were in initially?

After two days, 8962 chickens are expected to be in North field.
How many of these 8962 chickens are expected to have been in North field initially?

Of the chickens expected to be in East field after three days, what is the percentage of chickens that were also expected to be in East field after both one day and two days?
Round your answer to the nearest whole number.

Consider the following four matrices.

$A = \begin{bmatrix} 4 & 6 \\ -2 & -3 \end{bmatrix} \quad B = \begin{bmatrix} 4 \\ 2 \\ 8 \end{bmatrix} \quad C = \begin{bmatrix} 5 & 0 \\ 2 & 5 \\ 3 & 1 \end{bmatrix} \quad D = \begin{bmatrix} 3 & -2 & 5 \end{bmatrix}$

Which one of the following computations is defined?

A course requires students to pass two levels of study before they graduate.

Each level of study takes one year.

• Students must pass the first level ($L1$) before they move on to the second level ($L2$).
• Students must pass the second level before they become graduates ($G$).

From one year to the next, it is known that

• 80% of students pass the first level of study and progress to the second level of study
• 60% of students in the second level of study become graduates the following year.

Students who do not pass a level of study repeat it in the following year. No student leaves the course before graduating.

At the start of 2025, the numbers of students enrolled in $L1$ and $L2$ of the course and who have graduated from the course are given by the recurrence relation

$S_0 = \begin{bmatrix} 1000 \\ 0 \\ 0 \end{bmatrix} \begin{matrix} L1 \\ L2, \\ G \end{matrix} \quad S_{n+1} = TS_n + \begin{bmatrix} k \\ 0 \\ 0 \end{bmatrix}$

where

• $T$ is the transition matrix showing the expected proportion of students moving between states $L1$, $L2$ and $G$ from one year to the next
• $S_n$ is the state matrix $n$ years after the start of 2025.

To ensure that there are always students studying the course, new students are enrolled into the first level every year so that the number of students studying the first level remains at 1000.

$k$ is the number of new students enrolled at the first level each year.

At the start of which year will the expected number of students who have graduated from the course first exceed 5000?

Matrix $P$ is a permutation matrix and matrix $Q$ is a column matrix.

$P = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} \quad Q = \begin{bmatrix} t \\ e \\ a \\ m \\ s \end{bmatrix}$

When $Q$ is multiplied by $P$, which three letters change position?

Use the information in matrix $F$ to complete the table below.

There is an additional fee for each airfare.

One-way flights have an additional fee of ＄30 and return flights have an additional fee of ＄60.

The cost with the additional fee can be calculated as follows

Write down matrix $A$ in the space below.

$A =$

There will also be a 2% payment processing fee.

The total amount to be paid per flight, including this fee, is given by the following calculation

Determine the value of $p$.

Airline $B$ has optional upgrades as shown in the table below. Prices are per person per one-way flight.

The three friends will select upgrades for their flights.

The total cost of the upgrades is ＄130.

The following calculation is performed to determine the total cost of all their upgrades.

Where $U$ is a matrix representing the total number of each type of upgrade selected by the three friends.

Write down matrix $U$ in the space below.

$U =$

A shop sells mattresses.

The mattresses come in four sizes, single ($S$), double ($D$), queen ($Q$) and king ($K$).

The mattresses come in two comfort levels, lush ($L$) and firm ($F$).

Matrix $B$ below shows the number of each type of mattress sold in February 2025.

$B = \begin{bmatrix} 45 & 15 & 87 & 12 \\ 33 & 12 & 69 & 18 \end{bmatrix} \begin{matrix} L \\ F \end{matrix}$
with columns $S, D, Q, K$.

Matrix $C$ gives the predicted number of each type of mattress that will be sold during February 2026.

$C = \begin{bmatrix} 60 & 20 & 116 & 16 \\ 44 & 16 & 92 & 24 \end{bmatrix} \begin{matrix} L \\ F \end{matrix}$
with columns $S, D, Q, K$.

Matrix $C$ could have been calculated using which one of the following equations?

Matrix $K$ is a square matrix.

Multiplying $K$ by its inverse will result in

Consider the matrix $E$ where

$E = \begin{bmatrix} m & -9 \\ 4 & n \end{bmatrix}$

For the inverse of $E$ to exist, the values of $m$ and $n$, respectively, cannot be