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PARADE COLLEGE

Student Name: _______________________ Teacher Name: ___________________

FURTHER MATHEMATICS Unit 4

SAC 4-Matrices

Number of questions to be answered

Number of

marks

awarded

Number of marks

available

Section A: Multiple choice (5)

5

Section B: Short answer questions (8)

75

Total

80

Conditions and restrictions

Students are permitted to bring into the room for this task: One bound resource (text book

or work book), Graphics calculator and/or scientific calculator, pens, pencils, highlighters,

erasers, sharpeners and rulers.

Students are NOT permitted to bring into the room for this task: blank sheets of paper

and/or white out liquid/tape.

Test conditions – no form of communication permitted.

Instructions

Print your name in the space provided on the top of the front page.

Answer all questions in this booklet in the spaces provided.

If a question is worth two or more marks you must show working.

Unless otherwise stated, dollar values must be given to the nearest cent (2 decimal places).

Students are NOT permitted to bring mobile phones and/or any other unauthorised

electronic communication devices into the room for this task.

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Section A: Multiple Choice Section

Q1 If

3 1

2 6A

and 0

2 2

xB

and BA is equal to

44

23, the value of x is:

A 3

B 0

C –1

D –3

E 2

Q2 Which pair of matrices can be multiplied in the following order?

A 𝐴 = [2 4 2] and 𝐵 = [−3 − 1 3]

B 𝐴 = [2 4 2] and 𝐶 = [−3 − 1 3 4 6 3

]

C 𝐶 = [−3 − 1 3 4 6 3

] and D= [3 84 6

]

D 𝐶 = [

−3 − 1 3 4 6 3

] and 𝐸 = [3

−15

]

E D= [

3 84 6

] and 𝐸 = [3

−15

]

Q3

If 2 1

0 1A

then 2A is equal to:

A

10

14 D

2 1

0 1

B 4 2

0 2

E 1 1

0 2

C

10

14

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Q4

The transition matrix that can be used to represent the information in the diagram above is:

From From From

A To:

A B

A 0.75 0.80

B 0.25 0.20

B To:

A B

A 0.75 0.25

B 0.80 0.20

C To:

A B

A 0.25 0.80

B 0.75 0.20

From From

D To:

A B

A 0.75 0.20

B 0.25 0.80

E To:

A B

A 0.75 0.80

B 0.20 0.25

Q5

If 𝐴 = [6 3

−8 4], then the determinant is

A -1

B 15

C 0

D 48

E 42

B

65%

35%

90%

10%

0.65 0.90

0.35 0.10

0.65 0.35

0.90 0.10

0.35 0.90

0.65 0.10

0.65 0.10

0.35 0.90

0.65 0.90

0.30 0.15

A

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Section B: Short Answer Section

Question 1 (8 marks)

Peter bought tickets for an upcoming Taylor Swift concert. There are two types of tickets available,

adult, a, and concession, c. He bought tickets in bulk and does not know the individual prices.

Peter knows that he bought 4 adult tickets and 2 concession tickets for $ 600. He finds out that one of his

friends bought 3 adult and one concession ticket for $ 400.

a) Express the information above as two simultaneous equations in terms of a and c.

(2 marks)

b) Write the system of simultaneous equations in matrix form.

(2 marks)

c) For the square matrix in part (b), find

i) the determinant

(1 mark)

ii) the inverse

(1 mark)

d) Solve the matrix equation from (b) to find the price of each ticket.

(2 marks)

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Question 2 (6 marks)

If a mum is late picking up her child from school (L) on any given weekday, there is a 5% probability

that mum will be late on the next weekday. The mum is on time (O) one day, there is a 25% chance that

she will be late the next day.

a) Complete the following transition matrix that describes this situation:

today

O L

O next day

L (2 marks)

In a given week, the mum arrives on time on Monday, calculate:

b) The probability that the mum will be on time on the Wednesday (of the same week). The matrix that

describes the mum being on time on Monday is 1

0

(2 marks)

c) Given that the mum will be late on the Monday find the long term probability that mum will be late by

finding the steady state matrix. (Answer to two decimal places)

(2 marks)

T

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Question 3 (12 marks)

Mr Stark is buying some clothing items online. He has found three different online shops, although each

shop is from a different country and he would like to convert the prices to Australian Dollars ($AUD).

He likes two items, some shoes, s, and a wallet, w. He has found the same product at three stores. The

prices are listed below:

Shoes Wallet

UK £ 130 £ 370

USA $ US 250 $ US 490

Malta €200 €420

a) Write the above table as a clearly labelled a 3 x 2 matrix.

P

(2 marks)

In order covert each amount to $AUD, Mr Stark multiplies P by a new matrix, C, to come up with the

converted prices.

2.18 0 0

0 1.42 0

0 0 1.6

C

b) Explain why the product CP exists, but the product PC does not exist.

(2 marks)

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c) Calculate CP, and write the product, A = C x P, below (round to the nearest dollar).

A =

(2 marks)

d) Explain what the matrix A represents.

(1 mark)

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e) Mr Stark wishes to compare the cost of some of the items as well. He then purchases 4 pairs of shoes

and 6 wallets.

Represent this information as a single matrix labelled as matrix, K where Mr Stark buys 4 pairs of shoes

and 6 wallets from each store.

Shoes

Wallets

(1 mark)

f) Calculate AK, and write the product, M = AK.

Represent your solution as a clearly labelled Matrix, M. Where M shows the cost in Australian dollars

of purchasing from each country.

M =

(2 mark)

g) Write down the calculation that helps you find the value 21m

(2 mark)

K

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Question 4 (15 marks)

The Benevento Soccer Club offers two types of tickets to their members. ‘A’ tickets are more expensive

with better views of the ground. While ‘B’ tickets are less expensive, but provide a better atmosphere for

the fans. Members are free to change between areas on a week to week basis. In Round 1, 10 300

members choose section ‘A’ tickets and 6 200 choose section ’B’ tickets.

Complete the initial state matrix 1S for this situation:

1S

1S

(1 mark)

Of the people who go to the A section this week 28 % said that they would go to the A section next

round and the remainder said that they would like to go to the B section. Of the people who initially go to

section B in any round, 41% said they would go to the B section next week and the remainder wanted to

go to A.

b) Construct the transition matrix (T) for this situation

this week

A B

A next week

B

(2 marks)

c) Calculate the number of people expected to go to the A section in Round 2, ie 2S (round to the nearest

person).

(2 marks)

T

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d) Calculate the number of people expected to go to section A in Round 3, ie 3S

3S =

( 2 marks )

e) The administration has recorded that the numbers of people going to each section from week to week.

They believe that the number of people requiring each option will steady over time if the trends continue.

Use two series of matrix products in order to determine the values of the steady state of Section A & B

members.

(3 marks)

As it turns out, Benevento Soccer Club are having a surprisingly good year. After the fifteenth round, 15S ,

they begin to allow a certain number of new supporters to each game. They revise their model based on

the increased number of fans to each section each week.

11

1

520

420n nS T S

f) Given that there are 7431 supporters in Section A and 9069 supporters in Section B in Round 15,

ie 15

7431

9069S

Calculate S16, S17 and S18 (round to the nearest person)

16S =

17S

18S

(3 marks)

g) In which round will we see the total number of supporters exceed 20 000? Show how you came to

this conclusion

(2 marks)

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Question 5 (14 marks)

In a particular town in Victoria Vodaphone sold 8000 mobile phones in 2014 while the other provider

(O) sold 2000. Vodaphone sold 4000 Apple IPhones (A), 3000 Samsungs (S) and 1000 LGs (LG).

a) Establish an initial state matrix that shows the number of phones sold by Vodaphone and the other

provider.

A

S

LG

O

(2 marks)

Assume that all customers buy a mobile phone every year. They either upgrade their existing phone,

change brands or go to another provider (not Vodaphone).

47% of customers who purchased an Apple will stay with Apple, 25% will switch to Samsung and 16%

will switch to another provider.

54% of Samsung customers will stay with Samsung, 9% will switch to LG and 33% will switch to Apple.

The remainder switch to another provider.

48% of LG customers will stay with LG, 15% will buy their phones from Apple and 32% will switch to

Samsung. The remainder switch to another provider.

0s

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67% of non-Vodaphone customers will stay with their provider, others will join Vodaphone. Of those

that join 19% will switch to Apple and 6% will switch to LG. The remainder will switch to Samsung.

b) Write down the transition matrix, T that will represent this situation. Express T as a clearly labelled

4 x 4 matrix. A = Apple S = Samsung LG = LG and O = other

this year

A S LG O

A

S next year

LG

O

(4 marks)

c) Find the number of each brand of phone that were sold by Vodaphone in 2015 and the number sold by

the other provider. Give your answer to the nearest whole number.

(2 marks)

T

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d) Find the number of sales of each type of phone in 2016. Give answers rounded to the nearest whole

number.

(2 marks)

e) Find the steady state matrix for the number of sales of each brand by Vodaphone and the number that

use the other provider. Give your answer to the nearest whole number.

(2 marks)

f) Of the initial 10 000 people in the town buying phones what percentage in the long run will buy from

Vodaphone?

(2 marks)

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Question 6 (6 marks)

To ease congestion on the city loop, The Victorian Government will be building tunnels through the

CBD. Currently underway is the construction of a tunnel that will join Flinders St to Melbourne

Central Station.

The network matrix W is used to represent the information in this diagram.

From

Flin Par 𝑀𝐶 𝐹𝑙𝑎𝑔 𝑆𝐶

Flin

Par

MC To

Flag

SC

Flagstaff Melbourne

Central

Southern

Cross Parliament

Flinders St

0 1 1 0 1

1 0 1 0 0

1 1 0 1 0

0 0 1 0 1

1 0 0 1 0

W

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In matrix W

The 1 in row 2, column 1, for example, indicates that a train can travel directly from Flinders St to Parliament.

The 0 in row 2, column 5, for example, indicates that no trains travel directly from Southern

Cross to Parliament.

a) Find the sum of the elements in row 3 of the matrix W.

(1 mark)

b) In terms of the network described, what does the sum of the elements in row 3 of the matrix, W

represent?

(1 mark)

By analyzing Myki data, Metro discover that there is the varying demand for each journey during a

particularly busy time: This demand is represented by the matrix D2015, below.

From

Flin Par 𝑀𝐶 𝐹𝑙𝑎𝑔 𝑆𝐶

0 540 390 0 730

560 0 620 0 0

730 460 0 850 0

0 0 320 0 450

220 0 0 220 0

D

To

Flin

Par

MC

Flag

SC

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In order to match demand, Metro aim to provide the correct number of services. Each service can carry up

to 300 passengers. If the number of passengers exceeds 300, Metro will supply an extra service, and so

on until all passengers are accounted for.

c) Complete the matrix S2015 below, indicating the minimum number of services that Metro need to

supply for each leg of the journey.

from

Flin Par 𝑀𝐶 𝐹𝑙𝑎g 𝑆

2015

0 _ _ 0 _

_ 0 _ 0 0

_ _ 0 _ 0

0 0 _ 0 _

_ 0 0 _ 0

S

To

(2 marks)

Flin

Par

MC

Flag

SC

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Forecasts suggest that, as the population of the city increases, another link will need to be built, i.e. a

link between Flinders St and Flagstaff.

d) Complete the new network matrix below if a link between Flinders St and Flagstaff is

constructed.

Flin Par 𝑀𝐶 𝐹𝑙𝑎g 𝑆C

0 1 _ _ _

1 0 _ _ _

_ _ 0 _ _

_ _ _ 0 _

_ _ _ _ 0

N

(2 marks)

Flin

Par

MC

Flag

SC

Flagstaff Melbourne Central

Southern

Cross Parliament

Flinders St

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Question 7 (7 marks)

a) Below is a network diagram that displays the results of a tennis tournament between 5 individuals:

a) In the space below fill in the missing spaces that would indicate the one step domination links

between the individuals

Matrix D =

A B C D E

_ _ _ _ _

_ _ _ _ _

_ _ _ _ _

_ _ _ _ _

_ _ _ _ _

A

C

E

D

B

A

B

C

D

E

(2 marks)

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b) Find 2D and explain what the significance of the

2D matrix is.

A B C D E

_ _ _ _ _

_ _ _ _ _

_ _ _ _ _

_ _ _ _ _

_ _ _ _ _

A

B

C

D

E

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(2 marks)

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

(1 mark)

c) If the matrix T is defined as 2T D D use this relationship to explain who the

best tennis player is in the tournament.

(2 marks)

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Question 8 ( 7 marks )

In four small country towns (A, B, C and D) natural gas is stored for emergencies

during colder months. Gas can also be pumped from one town to another if

necessary as shown in the matrix M below:

Gas to

A B C D

A

B

Gas from M =

C

D

Note: elements in the matrix above row 1 column 4 means that town A can pump gas

to town D.

0 1 0 1

1 0 1 0

1 0 0 1

0 1 0 0

    

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a) Use the nodes below in order to show how the gas pumping operates as a

network diagram.

( 3 marks )

b) Find matrix 2P M M

( 1 mark )

A

C

B

D

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c) Why are the diagonal elements not useful in this context?

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

( 1 mark )

d) Using your result from part (b) above which town is most at risk of having the least

gas during a cold winter? Explain your answer.

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

(2 marks )

END of EXAMINATION