Matrices

[Maximum mark: 6]

The following transition diagram reflects the proportions of customers that Qatar Airways loses to its competitor airlines each year, and vice versa.

1. Construct a transition matrix $\bm{T}$ with elements in decimal form. [2]
   

2. Interpret the meaning of the elements with values

   1. $0.15$

   2. $0.75$ [2]
      

Assume that the initial state of the market share is

.

3. Determine the market share of Qatar Airways after $5$ years. [2]
   

[Maximum mark: 5]

There are $225$ senior students studying Computer Science and $169$ senior students studying Mathematics at a university. According to an academic survey, $20$ % of these $\text{students}$ tell they will pursue a postgraduate degree, $8$ % will start a business, $52$ % will get employed, $15$ % will start freelancing and the remaining students will become $\text{research}$ assistants.

The column matrix

represents the number of senior students studying

Computer Science and Mathematics.

1. Write down a row matrix, $\bm{R}$, to represent the percentages, in decimal form, of senior students who will choose one of the five routes after graduation. [1]
   

2. Hence calculate the product $\bm{A} = \bm{SR}$. Give each element $a_{ij}$ of the matrix $\bm{A}$ correct to the nearest whole number. [1]
   

3. In the context of this problem, explain what the element $a_{15}$ means. [1]
   

The cost for textbooks per year for a computer science student is $\$1245$ and for a mathematics student is $\$889$.

4. Write down a matrix calculation that gives the total cost for textbooks paid by all the senior students studying Computer Science and Mathematics. [1]
   

5. Hence calculate the total cost for all the textbooks. Give your answer correct to the nearest dollar. [1]
   

[Maximum mark: 4]

Domino's Pizza owns two pizzerias at Asia Mall and Metrocity shopping centres. The number of Pacific Veggie, Pepperoni and Buffalo Chicken large pizzas sold during the last week at the two pizzerias is shown in the table below.

The selling price of each type of pizza is shown in the table below.

1. Write down a matrix multiplication that finds the total amount of income from sales of the three types of pizzas that each pizzeria generated during the last week. [2]
   

2. Hence find the total amount of income from sales of the three types of pizzas that each pizzeria generated during the last week. Give your answers correct to two decimal places. [2]
   

[Maximum mark: 6]

Data scientists, web designers and developers are paid according to an $\text{industry}$ standard. The total annual salary spend for three tech startups paying to the industry standard are summarised in the table below.

Let $x$, $y$ and $z$ represent the salaries, in thousand dollars, for data scientists, web designers and web developers respectively.

1. Write down a system of three linear equations in terms of $x$, $y$ and $z$
   that represent the information in the table above. [2]
   

2. Using matrices, solve the system of linear equations from part (a)
   to determine the salaries for the three roles. [2]
   

Data Quant is a tech startup that also pays to the industry standard and employs $10$ data scientists, $4$ web designers and $6$ web developers.

3. Calculate the exact value of the total salary bill for Data Quant. [2]
   

[Maximum mark: 6]

The Brown, Miller and Taylor families pay utility bills for their houses each month. The table below shows the amount of electricity, water and gas consumed during January by each family, and the total cost of the utilities.

Let $x$, $y$ and $z$ represent the prices, in dollars, for $1$ kWh of electricity, $1$ m$^3$ of water and $1$ m$^3$ of gas, respectively.

1. Write down a system of three linear equations in terms of $x$, $y$ and $z$
   that represents the information in the table above. [2]
   

2. Using matrices, find the price for each of the utility. [2]
   

The Smith family also pay utility bills each month. The table below shows the amount of electricity, water and gas consumed during January by the Smith family.

3. Calculate the total cost of the utilities for the Smiths. [2]
   

[Maximum mark: 6]

Taste of Home Magazine recommends using a combination of Cheddar, Brie and Swiss when putting together cheese boards for parties. The recommended total cheese board size for a party of $10$ - $15$ people is $1$ kilogram. The table below shows the weight, in hundred of grams, of each kind of cheese required to make one kilogram of cheese combination, and the cost of making each combination.

1. By setting up a system of linear equations and using matrices, find
   the price per kilogram of each type of cheese. [4]
   

John prepares a cheese board with proportion of each cheese type, in hundred grams, as shown in the table below.

2. Calculate the amount of money John spent on cheese. [2]
   

[Maximum mark: 6]

Three Internet Service Providers (ISPs) are available in a small town. During the year, ISP A is expected to retain $85$ % of its customers; $10$ % will be lost to ISP B and $5$ % to ISP C. ISP B is expected to retain $80$ % of its customers; $10$ % will be lost to each of the other two ISPs. ISP C is expected to retain $75$ % of its customers; $15$ % will be lost to ISP A and $10$ % to ISP B.

1. Write down a transition matrix that describes the exchange of market shares between the three ISPs during the year. [2]
   

The current market share held by ISP A is $0.2$, by ISP B is $0.3$ and by ISP C is $0.5$.

2. Find the market share held by each ISP after one year. [2]
   

3. Find the market share held by each ISP after five years if the same trend of market share exchanges between the three ISPs continues. [2]
   

[Maximum mark: 8]

Two competing radio stations, station A and station B, each have $50$ % of the listener market at some point in time. Over each one-year period, station A $\text{manages}$ to take away $15$ % of station B's share, and station B manages to take away $10$ % of station A's share.

1. Write down a transition matrix that describes the exchange of market shares between the two stations over each one-year period. [1]
   

2. Find the market share held by each station after one year. [2]
   

3. Write down the market shares of stations A and B over a five-year period. [2]
   

4. Find the market share held by each station in the long term if the same trend of market share exchanges between the two stations continues indefinitely. [3]
   

[Maximum mark: 6]

When a coin is thrown from the top of a skyscraper, its height above the ground after $t$ seconds is given by $s(t) = at^2 + bt + c$, where $a,b,c \in \mathbb{R}$ and $s(t)$ is measured in metres. After $1$ second, the coin is $179.3$ m above the ground; after $2$ seconds, $188.2$ m; after $6$ seconds, $159.8$ m.

1. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Hence find the values of $a$, $b$ and $c$. [3]
      

2. Find the height of the skyscraper. [1]
   

3. Find the time it takes for the coin to hit the ground. [2]
   

[Maximum mark: 8]

Austin allocates a portion of his employment salary each month to investing and invests this money into two stock funds: A and B. He adjusts his investment portfolio each month according to the following transition diagram.

1. Construct a transition matrix $\bm{T}$ with elements in decimal form. [2]
   

2. Interpret the meaning of the elements with values

   1. $0.1$

   2. $0.7$ [2]
      

The initial state of his investment portfolio is $100\%$ in stock fund B.

3. 1. Find the investment proportion in stock fund A after $3$ months.

   2. Determine the long term steady state proportion of his investment between the two stock funds. [4]
      

[Maximum mark: 6]

When a ball is thrown from the top of a tall building, its height above the ground after $t$ seconds is given by
$s(t) = at^2 + bt + c\,,$ where $a,b,c \in \mathbb{R}$ and $s(t)$ is measured in metres. After $1$ second, the ball is $84.3$ m above the ground; after $2$ seconds, $93.9$ m; after $8$ seconds, $42.3$ m.

1. 1. Write down a system of three linear equations in terms of $a$, $b$ and $c$.

   2. Use matrices to find the values of $a$, $b$ and $c$. [3]
      

2. Find the height of the building. [1]
   

3. Find the time it takes for the ball to hit the ground. [2]
   

[Maximum mark: 6]

The vegetables sold at supermarkets in a town are supplied by three major retail suppliers: A, B and C. According to an analysis report, supplier A retains $80$ % of their customers each year and lose $15$ % to supplier B and $5$ % to supplier C. Meanwhile, supplier B retains $70$ % of their customers each year and lose $20$ % to supplier A and $10$ % to supplier C. Supplier C retains $75$ % of their customers each year and lose $10$ % to supplier A and $15$ % to supplier B.

The report also shows that suppliers A, B and C currently hold a market share of $50$ %, $25$ % and $25$ %, respectively.

1. Find the market share held by each supplier after three years. [4]
   

2. Determine the steady state market share held by each supplier if the same trend remains unchanged. [2]
   

[Maximum mark: 5]

Laura creates a list of her favorite songs that includes three genres: Jazz, Slow Rock and Country. After her current song ends she randomly selects the next song and the probabilities of genre of the next song are outlined in the following table.

Laura starts her day with a Slow Rock song and is now listening to her fourth song.

1. Determine the genre of music she is currently most likely listening to. [3]
   

2. Determine which genre of music she listens to most over the long term. [2]
   

[Maximum mark: 6]

A discrete dynamical system is described by the following transition matrix, $\bm{T}$,

The state of the system is defined by the proportions of population with a particular characteristic.

1. Use the characteristic polynomial of $\bm{T}$ to find its eigenvalues. [2]
   

2. Find the corresponding eigenvectors of $\bm{T}$. [2]
   

3. Hence find the steady state matrix $\bm{s}$ of the system. [2]
   

[Maximum mark: 14]

A biologist conducts an experiment to study the pollination preference of bumblebees' on different floral types. In a flight cage, $240$ bumblebees are free to choose between two species of floral: A. majus striatum or A. majus pseudomajus. The changes of pollination behaviors of these bumblebees after every minute are reflected in the following table.

Initially, $150$ bumblebees choose A. majus striatum and $90$ bumblebees choose A. majus pseudomajus.

1. Write down the initial state $\bm{s}_0$ and the transition matrix $\bm{T}$. [2]
   

2. Determine $\bm{Ts}_0$ and interpret the result. [2]
   

3. Find the eigenvalues and corresponding eigenvectors of $\bm{T}$. [4]
   

4. 1. Write an expression for the number of bumblebees choosing to pollinate on A. majus pseudomajus after $n$ minutes, $n \in \mathrm{N}$.

   2. Hence find the number of bumblebees choose to pollinate on A. majus pseudomajus in the long term. [6]
      

[Maximum mark: 14]

An information technology (IT) company offers paid travelling vacation to its $160$ employees every year. The employees can choose between travelling domestically or internationally. It is observed that $50$ % of the employees who choose to travel domestically one year, choose internationally the next year. Conversely, $30$ % of those who choose to travel internationally one year change to travel domestically the following year. For this year, $80$ employees chose travelling domestically and $80$ employees chose travelling internationally.

1. Write down the initial state $\bm{s}_0$ and the transition matrix $\bm{T}$. [2]
   

2. Determine $\bm{Ts}_0$ and interpret the result. [2]
   

3. Find the eigenvalues and corresponding eigenvectors of $\bm{T}$. [4]
   

4. 1. Write an expression for the number of employees who choose travelling internationally after $n$ years, $n \in \mathrm{N}$.

   2. Hence find the long term steady state number of employees to choose to travel internationally. [6]
      

[Maximum mark: 28]

Steve watches a TV show that has a small section dedicated to trivia quiz questions, where a randomly selected person from the audience is asked to answer 5 general knowledge questions in a row. The following table records the number of correct answers Steve obtains from watching 100 episodes of the program.

1. Find Steve's mean number of correct answers per show. [2]
   

Steve suspects that his number of correct answers can be modelled by a binomial distribution with $n=5$, and decides to carry out a $\chi^2$ goodness of fit test.

2. Using the data from the table, find the estimated value of $p$ for the binomial model.[1]
   

3. 1. Use the binomial model to find the probability that Steve gets all 5 questions incorrect. Express your answer to two significant figures.

   2. Find the expected frequency for zero correct answers.[3]
      

As some expected frequencies are less than $5$, Steve combines two rows in the table to obtain the following updated table.

4. Steve uses this table to carry out a $\chi^2$ goodness of fit test to test the hypothesis that the data follow a binomial distribution with $n=5$ at the $10\%$ significance level. For this test, state:

   1. the null hypothesis;

   2. the number of degrees of freedom;

   3. the $p$ value;

   4. the conclusion, justifying your answer.[6]
      

5. Using the binomial model, find the probability that during the next show, Steve answers all five questions correctly.[2]
   

Steve's friend Tony considers it might be better to model the problem by using states, where R means a correct answer, and W is an incorrect answer. To simplify the model, Tony proposes a transition of only two states with frequencies shown in the following table.

6. 1. Find the probability, in the form $\dfrac{p}{q}$, that Steve gets a question incorrect given that he correctly answered the previous question.

   2. Write down the transition matrix, $\bm{T}$, for this model.[3]
      

7. 1. Show that the characteristic polynomial for the transition matrix can be expressed as $39\lambda^2-47\lambda + 8 = 0$.

   2. Hence, or otherwise, find the eigenvalues of $\bm{T}$.

   3. Find the eigenvectors of $\bm{T}$.[7]
      

Steve is leaving his home country for a working vacation and won't see the show for at least 6 months. When he returns, he will watch the first episode of the show and attempt to get all five questions correct. Tony claims that, according to the transition model, the probability of this occurring is approximately 10%.

8. Determine whether Tony's claim is correct.[4]
   

[Maximum mark: 17]

Two grocery stores, store A and store B, serve in a small city. Each year, store A keeps $30$ % of its customers while $70$ % of them switch to store B. Store B keeps $\text{\(60\)\hspace{0.05em}\%}$ of its customers while $40$ % of them switch to store A.

1. Write down a transition matrix $\bm{T}$ representing the proportions of the $\text{customers}$ moving between the two stores. [2]
   

At the end of $2019$, store A had $8360$ customers while store B had $6820$ customers.

2. Find the distribution of the customers between the two stores after two years.[2]
   

3. 1. Show that the eigenvalues of $\bm{T}$ are $\lambda_1 = 1$ and $\lambda_2 = -0.1$.

   2. Find a corresponding eigenvector for each eigenvalue from part (c) (i).

   3. Hence express $\bm{T}$ in the form $\bm{T} = \bm{PDP}^{-1}$. [6]
      

4. Show that

   

   , where $n \in \mathbb{Z}^+$. [2]
   

5. Hence find an expression for the number of customers buying groceries from store A after $n$ years, where $n \in \mathbb{Z}^+$ [3]
   

6. Verify your formula by finding the number of customers buying groceries from store A after two years and comparing with the value found in part (b). [1]
   

7. Write down the long-term number of customers buying groceries from store A.[1]
   

[Maximum mark: 14]

Zoologists have been collecting data about the migration habits of a particular species of mammals in two regions; region X and region Y. Each year $\text{\(30\)\hspace{0.05em}\%}$ of the mammals move from region X to region Y and $15$ % of the mammals move from region Y to region X. Assume that there are no mammal movements to or from any other neighboring regions.

1. Write down a transition matrix $\bm{T}$ representing the movements between the two regions in a particular year. [2]
   

2. 1. Find the eigenvalues of $\bm{T}$.

   2. Find a corresponding eigenvector for each eigenvalue of $\bm{T}$.

   3. Hence write down matrices $\bm{P}$ and $\bm{D}$ such that $\bm{T} = \bm{PDP}^{-1}$. [6]
      

Initially region X had $12\hspace{0.15em}600$ and region Y had $16\hspace{0.15em}200$ of these mammals.

3. Find an expression for the number of mammals living in region Y after
   $n$ years, where $n \in \mathbb{Z}^+$. [5]
   

4. Hence write down the long-term number of mammals living in region Y. [1]
   

[Maximum mark: 14]

A city has two major security guard companies, company A and company B. Each year, $15$ % of customers using company A move to company B and $5$ % of the customers using company B move to company A. All additional losses and gains of customers by the companies can be ignored.

1. Write down a transition matrix $\bm{T}$ representing the movements between the two companies in a particular year. [2]
   

2. 1. Find the eigenvalues and corresponding eigenvectors of $\bm{T}$.

   2. Hence write down matrices $\bm{P}$ and $\bm{D}$ such that $\bm{T} = \bm{PDP}^{-1}$. [6]
      

Initially company A and company B both have $3600$ customers.

3. Find an expression for the number of customers company A has after $n$ years, where $n\in\mathbb{Z}$. [5]
   

4. Hence write down the number of customers that company A
   can expect to have in the long term. [1]