IBMaths: AI HLDPTopic Questions1. Number & Algebra1.7 Matrices

Matrices (DP IB Maths: AI HL)

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1.7 Matrices

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1a
Sme Calculator4 marks

Consider the following matrices:

       bold italic A equals open parentheses table row 1 0 cell negative 2 end cell row cell negative 1 end cell a 3 end table close parentheses blank bold italic B equals open parentheses table row b 2 row 1 cell negative 1 end cell end table close parentheses blank bold italic C equals open parentheses table row 0 c cell negative 3 end cell end table close parentheses 

      bold italic D equals open parentheses table row 0 1 0 row 3 d 1 row 0 3 0 end table close parentheses blank bold italic E equals open parentheses table row cell negative 1 end cell 4 e row cell negative 2 end cell 1 0 end table close parentheses blank bold italic F equals open parentheses table row f row 2 row cell negative 1 end cell end table close parentheses blank

where a comma blank b comma blank c comma blank d comma blank e comma blank f element of straight real numbers are constants.

Find each of the following matrix sums or differences, or if that is not possible then explain why:

(i)  bold italic A plus bold italic B                 (ii) bold italic A plus bold italic E             (iii) bold italic E minus 3 bold italic A                  

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1b
Sme Calculator5 marks

Find each of the following matrix products, or if that is not possible then explain why:

(i) bold italic B bold italic E    (ii) bold italic E bold italic B    (iii) bold italic C bold italic D    (iv) bold italic F bold italic C   (v) bold italic A bold italic E  

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1c
Sme Calculator3 marks

List any other matrix products of two of the above matrices that it would be possible to find, other than the ones included in part (b).  You do not need to find the products, but do specify what the order of each of those products would be.

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1d
Sme Calculator3 marks

Find each of the following:

(i) bold italic B bold left parenthesis bold italic A bold plus bold italic E bold right parenthesis            (ii) bold italic B bold italic A bold plus bold italic B bold italic E  

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2a
Sme Calculator7 marks

Consider the matrices:

             bold italic A equals open parentheses table row 2 cell negative 1 end cell row 4 cell negative 1 end cell end table close parentheses blank bold italic B equals open parentheses table row 1 1 row cell negative 4 end cell k end table close parentheses

where k element of straight real numbers is a constant.

Let bold italic I be the 2 cross times 2 identity matrix, and let 0 be the 2 cross times 2 spacezero matrix.

Find the following

(i) bold italic B to the power of bold 2    (ii) bold italic A bold minus bold 2 bold italic I    (iii) bold italic A bold italic B   (iv) bold italic B bold italic A   (v) bold italic A to the power of bold minus bold 1 end exponent bold italic B  

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2b
Sme Calculator4 marks

Find the following: 

(i)  det space bold italic A    (ii) det space bold italic A to the power of negative 1 end exponent    (iii) det space bold italic B   (iv) det space bold italic A bold italic B                 

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2c
Sme Calculator4 marks

bold italic C and bold italic D are matrices such that  bold italic A plus bold italic C bold equals bold 0  and  bold italic B to the power of negative 1 end exponent plus bold italic D equals bold italic   bold 0

 
(i)
 Write down matrix bold italic C. 
   
(ii)
 Find matrix bold italic D.

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3a
Sme Calculator3 marks

Consider the matrix

             bold italic A equals open parentheses table row 3 cell negative 8 end cell row p 7 end table close parentheses

where space p element of straight real numbers is a constant.

Given that det space bold italic A equals negative 3, find the value of space p.

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3b
Sme Calculator4 marks

Consider the two matrices

             bold italic B equals open parentheses table row 4 q row cell negative 1 end cell 3 end table close parentheses blank and          blank bold italic C equals open parentheses table row r 1 row 1 3 end table close parentheses

where q comma blank r element of straight real numbers are constants.

Given that bold italic B bold italic C equals bold italic C bold italic B, find the values of q and r.

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4a
Sme Calculator2 marks

Consider the matricesbold space bold italic P equals open parentheses table row 2 cell negative 1 end cell row 3 1 end table close parentheses  and bold italic D equals open parentheses table row 1 0 row 0 cell 0.8 end cell end table close parentheses.

Find

(i)     the determinant of bold italic P
 
(ii)    bold italic P to the power of negative 1 end exponent.

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4b
Sme Calculator3 marks

Consider the matrix product bold italic P bold italic D bold italic P to the power of negative 1 end exponent.

(i)
Show that open parentheses bold italic P bold italic D bold italic P to the power of negative 1 end exponent close parentheses squared equals bold italic P bold italic D squared bold italic P to the power of negative 1 end exponent and open parentheses bold italic P bold italic D bold italic P to the power of negative 1 end exponent close parentheses cubed equals bold italic P bold italic D cubed bold italic P to the power of negative 1 end exponent.

(ii)
Use the results of part (b)(i) to suggest an expression for open parentheses bold italic P bold italic D bold italic P to the power of negative 1 end exponent close parentheses to the power of n in terms of bold italic P, bold italic D and bold italic P to the power of negative 1 end exponent.

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4c
Sme Calculator1 mark

The transition matrix of a dynamic system is bold italic T equals open parentheses table row cell 0.88 end cell cell 0.08 end cell row cell 0.12 end cell cell 0.92 end cell end table close parentheses.

Find  bold italic T to the power of bold 5.

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4d
Sme Calculator4 marks
(i)
Show that bold italic T equals bold italic P bold italic D bold italic P to the power of negative 1 end exponent.

(ii)
Hence use the answer to part (c) to confirm the validity of your expression from part (b)(ii) for n equals 5.

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5a
Sme Calculator1 mark

A family are buying burgers for dinner and wish to order three veggie burgers, one beef burger and two chicken burgers.

From burger store A three veggie burgers would cost a total of $15.45, one beef burger would cost $6.15, and 2 chicken burgers would cost a total of $11.90.

Write down

(i)
a row matrix, bold italic Q, to represent the quantities of each type of burger that the family wishes to purchase

(ii)
a column matrix, bold italic P subscript A, to represent the cost of each type of burger at store A.

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5b
Sme Calculator3 marks

Burger store B sells their veggie burger, beef burger and chicken burger for $4.75, $5.85 and $5.50, respectively.

(i)
Write down a column matrix, bold italic P subscript B, to represent the cost of each type of burger at store B.

(ii)
Hence write down a cost matrix, bold italic C, to represent the costs for both stores.

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5c
Sme Calculator2 marks

By first calculating the matrix bold italic Q bold italic C, compare the total cost of the family’s dinner at stores A and B.

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6a
Sme Calculator1 mark

A café is looking to hire a duty manager, two baristas, a dishwasher, and three waiters.  They decide to advertise the jobs on social media, as well as using a hiring agency.  They receive 123 applications from advertising the job on social media and 57 applications from the hiring agency.

Write down a column matrix, bold italic C, to represent the number of applications they received from advertising the job on social media and from using the hiring agency.

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6b
Sme Calculator1 mark

Overall, 22% of the applicants applied for the duty manager job, 28% applied for the barista job, 18% applied for the dishwasher job, and the rest applied for the waiter job.  Note that every applicant was only allowed to apply for one of the available jobs.

Write down a row matrix, bold italic R, to represent the percentages of the applicants that applied for each of the different jobs.

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6c
Sme Calculator3 marks
(i)
Calculate the product bold italic P equals bold italic C bold italic R.

(ii)
Use the elements of the matrix bold italic P to work out the total number of applicants for each of the positions, giving your answers to the nearest integer.

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6d
Sme Calculator2 marks

Once the café has selected the right candidate for each job, each new employee will work a total of 40 hours per week.  The hourly wage for the duty manager and barista jobs is $20.00 per hour, and the hourly wage for the dishwasher and waiter jobs is $17.25.

Calculate the café’s weekly wage expenses for the new employees.

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7a
Sme Calculator2 marks

Amanda is a landlord who has new sets of tenants moving into three new unfurnished homes.  Amanda receives a special deal at a small local furniture store and so she offers her tenants that she will buy some tables, chairs, and/or sofas on their behalf, given that they reimburse her for the cost.  The quantities of different items ordered for each house are shown in the table below.

 

Tables

Chairs

Sofas

House 1

2

5

2

House 2

1

3

0

House 3

1

4

1

(i)
Write down a 3 cross times 3 matrix, bold italic F, to represent the furniture orders for the three houses.

(ii)
Write down a 1 cross times 3 row matrix, bold italic S, to represent Amanda’s total shopping list at the furniture store.

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7b
Sme Calculator5 marks

The price Amanda pays for each item is shown in the table below.

 

Tables

Chairs

Sofas

Price

$72.00

$14.50

$47.50

(i)
By performing an appropriate matrix multiplication with matrix bold italic F, find the total amount owed to Amanda by each house.

(ii)
By performing an appropriate matrix multiplication with matrix bold italic S, find the total amount paid by Amanda to the furniture store.

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8a
Sme Calculator2 marks

The bus fare a person pays in a city is dependent on whether the person is a student, an adult or a pensioner.

The total amount taken in on a particular day by three different buses, along with the numbers of each type of fare paid, are shown in the table below.

 

Student

Adult

Pensioner

Total

Bus A

91

82

13

$348

Bus B

102

80

4

$355

Bus C

71

54

11

$247

 

Let s comma space a andspace p represent the amount paid by a student, an adult, and a pensioner respectively.

Write down a system of three linear equations in terms of s comma space a andspace p that represent the information shown in the table above.

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8b
Sme Calculator2 marks

Find the values of s comma space a andspace p using appropriate matrices and matrix inverses.

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8c
Sme Calculator2 marks

Bus D finished the day having sold 112 student tickets, 91 adult tickets and 22 pensioner tickets.

Calculate the total amount taken in by Bus D. Give your answer to 2 decimal places.

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9a
Sme Calculator3 marks

The number of times a gamer logged in to Call of Duty, FIFA, or Assassin’s Creed over 3 weeks is shown in the table below.

 

Call of Duty

FIFA

Assassin’s Creed

Week 1

5

4

3

Week 2

7

2

4

Week 3

3

5

6

The total number of hours the gamer spent playing each week is shown in the table below.

 

Week 1

Week 2

Week 3

Total hours

12.35

13.84

14.16

 

The gamer was never logged in to more than one game at the same time.

The gamer believes that, for each game, the average amount of time spent playing per   log-in session was consistent over the three weeks. 

Assuming that the gamer’s belief is true, use matrix multiplication to find the average number of hours and minutes per log-in session that the gamer spent playing each game.

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9b
Sme Calculator2 marks

Write down a system of linear equations that could be used to find the answers in part (a).

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10a
Sme Calculator2 marks

The graph of the quadratic function space f left parenthesis x right parenthesis equals a x squared plus b x plus c passes through the points left parenthesis negative 1 comma 11 right parenthesis comma space left parenthesis 3 comma negative 5 right parenthesis and left parenthesis 6 comma space 4 right parenthesis.

Show that a, b and c must satisfy the following system of linear equations: 

a minus b plus c equals 11

9 a plus 3 b plus c equals negative 5 

36 a plus 6 b plus c equals 4

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10b
Sme Calculator1 mark

Represent the system of equations in part (a) in matrix form.

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10c
Sme Calculator3 marks

Hence use a matrix method to find the values of a, b and c.

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11a
Sme Calculator2 marks

The graph of the function space f left parenthesis x right parenthesis equals a x squared plus b x plus c passes through the points left parenthesis negative 1 comma space 5 right parenthesis comma space left parenthesis 3 comma 1 right parenthesis spaceand left parenthesis 4 comma negative 5 right parenthesis.

Write down a system of linear equations that a comma space b and c must satisfy.

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11b
Sme Calculator4 marks

Hence use a matrix method to determine the values of a comma space b and c.

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12a
Sme Calculator4 marks

The amounts of wheat, soybeans and sugar produced by three different farms in a given week, along with the respective total revenues for each farm, are shown in the table below.

 

Wheat, kg

Soybeans, kg

Sugar, kg

Revenue, $

Farm A

820

532

535

835.54

Farm B

1210

641

274

948.75

Farm C

922

211

503

716.11

 

Let x comma space y and z represent the prices, in $/kg, for wheat, soybeans and sugar respectively.

(i)
Write down a system of linear equations that represents the information in the table above.

(ii)
Solve the system of linear equations using matrices.

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12b
Sme Calculator3 marks

In the same week, Farm D produced a fifth of the amount of wheat as Farm A, a quarter of the amount of soybeans as Farm B, and half the amount of sugar as Farm C.

Calculate the revenue made by Farm D from selling these crops. Give your answer correct to two decimal places.

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13a
Sme Calculator6 marks

Grace has decided that she wants to invest $10 000 split between three companies: company A, company B, and company C.  She creates three different portfolio options based on risk levels, and calculates what each option’s value would be today if the identical amounts had been invested one year ago.

 

Company A

Company B

Company C

Value

Safe

$1500

$8000

$500

$10,620.00

Middle

$2000

$6750

$1250

$10,827.50

Risky

$2500

$2500

$5000

$11,725.00

 

Use a matrix method to find the annual percentage return (i.e., the percentage increase or decrease of an investment in the company) for the previous year for each company.

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13b
Sme Calculator2 marks

Grace hears some good news about the growth of company A before she invests her money, and so she decides to put 78% of it into company A and split the rest evenly between company B and company C.  

Compared with the previous year, the annual percentage return for company A for the coming year is expected to increase by 26 percentage points (so if the previous year’s return was x%, then the return is expected to be left parenthesis x plus 26 right parenthesis% for the coming year).  For company B the return is expected to remain the same, while for company C it is expected to decrease by 8 percentage points.

Find the expected value of Grace’s investment at the end of the coming year.

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1a
Sme Calculator2 marks

Consider the following matrices:

bold italic A equals open parentheses table row x 7 row cell negative 4 end cell 3 end table close parentheses space space space space space space space space bold space bold italic B equals open parentheses table row 2 y row 4 8 end table close parentheses

bold italic C equals open parentheses table row 1 a row 4 8 end table close parentheses space space space space space space bold space bold space bold italic D equals open parentheses table row b 2 row cell negative 9 end cell cell negative 10 end cell end table close parentheses space space space space space space space space space space space space space bold italic E equals open parentheses table row cell negative 34 end cell 62 row cell negative 66 end cell cell negative 52 end cell end table close parentheses

bold italic M equals open parentheses table row 3 cell 2 c end cell row cell negative c end cell 6 end table close parentheses space space space space space space space space space space space space space bold italic N equals open parentheses table row 10 cell d plus 24 end cell row d 22 end table close parentheses

where a comma space b comma space c comma space d comma space x comma space y element of straight real numbers are constants.

Given that bold italic A plus q bold italic B equals z bold italic I,  find the values of x, y, z, and q.

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1b
Sme Calculator3 marks

Given that 1 half bold italic E equals r bold italic C plus s bold italic D   find the values of a, b, r, and s.

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1c
Sme Calculator3 marks

Given that e bold italic M minus f bold italic N equals bold italic I find the values of c, d, e, and f.

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2
Sme Calculator3 marks

Consider the two matrices

 bold italic A equals open parentheses table row 3 2 row 2 0 end table close parentheses

B equals open parentheses table row 13 6 row 6 q end table close parentheses

where q element of straight real numbers is a constant. 

Given that bold italic Aand bold italic B are commutative, find the value of q.

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3
Sme Calculator3 marks

Consider the matrix

bold italic M equals open parentheses table row 3 0 row 0 cell negative 2 end cell end table close parentheses 

Find an expression for bold italic M to the power of k .

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4
Sme Calculator8 marks

Consider the matrices:

 bold italic A equals open parentheses table row 2 a row 3 4 end table close parentheses space space space space space bold italic B equals open parentheses table row 1 3 4 row 3 cell negative 2 end cell b end table close parentheses space space space space space space bold space bold italic C equals open parentheses table row 1 2 3 row 4 c 5 row cell negative 5 end cell cell negative 3 end cell cell negative 2 end cell end table close parentheses

bold italic D equals open parentheses table row d 1 row 2 3 end table close parentheses space space space space bold space bold italic E equals open parentheses table row 3 2 cell negative 1 end cell row e cell negative 2 end cell 4 row 1 cell negative 3 end cell 2 end table close parentheses space space space space bold italic F equals open parentheses table row 2 f row cell negative 2 end cell cell negative 1 end cell row 3 2 end table close parentheses

where a comma space b comma space c comma space d space e comma space f element of straight real numbers are constants. 

Find the following products in terms of the appropriate constants.  If it is not possible to do so, explain why.

(i)
bold italic A bold italic B
(ii)
bold italic C bold italic A
(iii)
bold italic E bold italic B
(iv)
bold italic B bold italic E
(v)
bold italic E bold italic A bold plus bold italic F
(vi)
bold italic F stretchy left parenthesis C minus E stretchy right parenthesis
(vii)
open parentheses bold italic C bold minus bold italic E close parentheses bold italic F

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5
Sme Calculator5 marks

Consider the following matrices:

 bold italic M equals open parentheses table row a b row c d end table close parentheses space space bold space bold space bold space bold italic N equals open parentheses table row e f row g h end table close parentheses space space bold space bold space bold space bold italic P equals open parentheses table row i j row k l end table close parentheses 

Show that open parentheses bold italic M bold italic N close parentheses bold italic P equals bold italic M open parentheses bold italic N bold italic P close parentheses and state the name of this property.

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6a
Sme Calculator2 marks

For each of the following matrices, 

(i)     find the values of x for which bold italic M to the power of negative 1 end exponent does not exist, and 

(ii)    for the cases where bold italic M to the power of negative 1 end exponent does exist, find bold italic M to the power of negative 1 end exponent in terms of x.

bold italic M equals open parentheses table row x cell negative 2 end cell row 1 3 end table close parentheses

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6b
Sme Calculator2 marks

bold italic M equals open parentheses table row cell x minus 2 end cell 3 row 4 cell 2 x end cell end table close parentheses

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6c
Sme Calculator2 marks

bold italic M equals open parentheses table row cell x squared end cell cell x minus 1 end cell row cell 3 x end cell 5 end table close parentheses

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7a
Sme Calculator4 marks

A message is encoded using a matrix. Letters in the message are represented by numbers as given in the table below.

 

A

B

C

D

E

F

G

H

I

J

K

L

M

1

2

3

4

5

6

7

8

9

10

11

12

13

 

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

14

15

16

17

18

19

20

21

22

23

24

25

26

 

Messages are encoded by splitting the message into pairs of letters and writing in a 2 space cross times space n  matrix.  For example, “encode” becomes open parentheses table row straight E straight C straight D row straight N straight O straight E end table close parentheses or open parentheses table row 5 3 4 row 14 15 5 end table close parentheses. Then the message is multiplied, on the left, by an encryption matrix.

Here is a message which has been encoded using the encryption matrix open parentheses table row 3 2 row cell negative 1 end cell 4 end table close parentheses:

open parentheses table row 41 96 33 53 96 46 row cell negative 9 end cell 52 3 71 66 8 end table close parentheses 

Decode the message.

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7b
Sme Calculator1 mark

A spy (whose name cannot be given) notices that if the message has an odd number of letters, it will not completely fill a 2 cross times n matrix. The spy suggests putting the message into a 3 cross times n matrix if the number of letters in the message is a multiple of three.

Assuming the same encryption matrix is used as was used for the above message, explain the problem with the spy’s suggestion.

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8a
Sme Calculator2 marks

Using the properties of matrices, explain why the following misconceptions are incorrect.

“I know that open parentheses a plus b close parentheses open parentheses a minus b close parentheses equals a squared minus b squared,  therefore  open parentheses bold italic A plus bold italic B close parentheses open parentheses bold italic A minus bold italic B close parentheses equals bold italic A to the power of bold 2 minus bold italic B to the power of bold 2 must also be true if A and B are matrices.”

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8b
Sme Calculator2 marks

“I know that open parentheses a plus b close parentheses squared equals a squared plus 2 a b plus b squared,  therefore open parentheses bold italic A plus bold italic B close parentheses squared equals bold italic A to the power of bold 2 plus bold 2 bold italic A bold italic B plus bold italic B to the power of bold 2 must also be true if A  and B are matrices.”

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9a
Sme Calculator5 marks

In this question bold italic A and bold italic B are 2 cross times 2  matrices, k comma space p comma space q element of straight real numbers are constants, and n is a positive integer.

(i)
Show that open parentheses p bold italic A close parentheses open parentheses q bold italic B close parentheses equals p q open parentheses bold italic A bold italic B close parentheses.

(ii)
Hence show that open parentheses k bold italic A close parentheses to the power of n equals k to the power of n bold italic A to the power of n. 

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9b
Sme Calculator2 marks

A student claims it is always true that open parentheses bold italic A bold italic B close parentheses double apostrophe equals bold italic A double apostrophe bold italic B double apostrophe.

Either explain why the student’s claim is always true, or else show that it is not always true by providing a counterexample.

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10a
Sme Calculator3 marks

Consider the following matrices:

 bold italic A equals open parentheses table row 5 p row p 7 end table close parentheses space space space space bold space bold italic B equals open parentheses table row 11 q row q 3 end table close parentheses 

where p comma space q element of straight real numbers are constants, q greater than 0.

Given that bold italic Aand bold italic B are commutative, find q in terms of p.

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10b
Sme Calculator2 marks

Given that the determinant of bold italic Ais 26, find p and q.

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11a
Sme Calculator3 marks

A couple are planning their wedding reception and wish to buy a bow for every chair, with a matching tablecloth for each table. 

There are a total of 120 chairs and 15 tables, all of which need bows and tablecloths respectively. 

Company A charges £1.03 per chair bow and £14 per tablecloth.

Company B charges £0.85 per chair bow and £16 per tablecloth. 

By setting up one matrix equation that includes both companies, compare the overall prices that would be charged by the two companies.

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11b
Sme Calculator3 marks

Married friends of the couple recommend Company C, whom they used for their wedding. The friends can remember that they paid £13 per tablecloth, but cannot remember the price per chair bow.

Set up and solve a matrix equation to find the maximum price per chair bow that Company C could charge so as still to be cheaper overall than companies A and B.

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12a
Sme Calculator4 marks

Veterinarians, veterinary nurses, and animal care assistants are paid fixed salaries according to an industry standard. The totals of the annual payrolls for three veterinary practices that pay according to the industry standard are summarised in the table below.

 

Practice

Veterinarians

Veterinary Nurses

Animal Care Assistants

Total Salary Spend

Aspen Road Vets

3

5

2

$ 294000

Broadoak Way Vets

2

2

1

$ 158000

Cats n Dogs Vets

7

10

4

$ 634000

 

Using matrices, set up and solve a system of equations to find the fixed salaries that are paid for each of the three roles.

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12b
Sme Calculator2 marks

Vicky is setting up a new veterinary practice, and to help recruit staff she is planning to pay 5% above the industry standard for all job roles.  She uses the following matrix multiplication to help find the total cost of her staffing, where p comma space q and r represent the salaries for a veterinarian, a veterinary nurse, and an animal care assistant respectively:

 a cross times open parentheses table row 3 4 2 end table close parentheses cross times open parentheses table row p row q row r end table close parentheses= Total salary spend in thousands

(i)
Write down the value of the constant a that Vicky should use

(ii)
Interpret the meaning of the element ‘4’ in the row matrix.

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13a
Sme Calculator4 marks

Consider a curve with equation y equals a x cubed plus b x squared plus c x minus 8, where a, b, and c are real constants.  The graph passes through the points straight A open parentheses 2 comma space 116 close parentheses comma space straight B open parentheses negative 4 comma space minus 712 close parentheses and straight C open parentheses 3 comma space 394 close parentheses. 

(i)
Use a matrix method to find the values of a comma space band c.

(ii)
Hence sketch the curve.

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13b
Sme Calculator2 marks

Consider a second curve with equation a x to the power of 5 plus b x to the power of 4 plus c x cubed plus d x squared plus e x plus f,  where a, b, c, d, e and f are real constants with a not equal to 0. 

By considering the method used to solve part (a), suggest the number of coordinates that would need to be known to determine the values of all the constants in the equation of the second curve.

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1a
Sme Calculator4 marks

Consider the following matrices:

 bold italic A equals open parentheses table row a 4 5 row 1 2 cell 6 b end cell end table close parentheses space space space space space space space space space bold italic B equals open parentheses table row 1 3 end table close parentheses space space space space space space space space bold italic C equals open parentheses table row cell negative 2 end cell row cell negative 1 end cell row 0 end table close parentheses 

The matrix product open parentheses bold italic M bold italic N close parentheses bold italic P is calculated, where M, N, and P can each be any one of the matrices A, B or C.

Find the possible dimensions of the resulting products.

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1b
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The matrix sum M + N + P   is calculated, where M, N, and P can each be any one of the matrices A, B or C.

Find the possible sums that could result from such an addition.

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1c
Sme Calculator5 marks

It is given that open parentheses bold italic C bold italic B close parentheses bold italic A equals t open parentheses table row 2 4 cell 9.2 end cell row 1 2 cell 4.6 end cell row 0 0 0 end table close parentheses.

Find the values of a, b, and t.

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2a
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In this question A, B, and C are arbitrary square matrices.

Prove the following matrix results, stating any necessary assumptions:

(i)
If  bold italic A bold italic B equals bold italic C then bold italic B equals bold italic A to the power of negative 1 end exponent bold italic C 

(ii)
open parentheses bold italic A bold italic B close parentheses to the power of negative 1 end exponent equals bold italic B to the power of negative 1 end exponent bold italic A to the power of negative 1 end exponent 

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2b
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Using the result from part (a)(ii), simplify open parentheses bold italic A to the power of negative 1 end exponent bold italic B close parentheses to the power of negative 1 end exponent bold italic A to the power of negative 1 end exponent.

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3a
Sme Calculator5 marks

Consider the matrices:

 bold italic M equals open parentheses table row 3 2 row 2 0 end table close parentheses

bold italic N equals open parentheses table row a b row c d end table close parentheses 

where a comma space b comma space c comma space d element of straight real numbers are constants. 

Given that M and N are commutative, find an expression for N in terms of b and d only.

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3b
Sme Calculator4 marks

Given that the inverse matrix open parentheses bold italic M bold italic N close parentheses to the power of negative 1 end exponent exists 

(i)
determine a relationship that the constants a comma b comma space cand d must satisfy 

(ii)
find open parentheses bold italic M bold italic N close parentheses to the power of negative 1 end exponent in terms of a comma space b comma space c and d.

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4
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Given that

 bold italic A equals open parentheses table row 2 8 row 2 cell negative 4 end cell end table close parentheses

bold italic A to the power of negative 1 end exponent bold italic B bold italic A equals open parentheses table row 2 0 row 5 6 end table close parentheses

bold italic A bold italic B bold italic C equals 3 bold italic I

find matrices B and C.

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5a
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For any 2 cross times 2 matrices M, A or B:

Prove that det open parentheses k bold italic M close parentheses equals k squared det bold italic M,  where k is a real constant.

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5b
Sme Calculator5 marks

Prove that det open parentheses bold italic A bold italic B close parentheses equals det space bold italic A cross times det space bold italic B.

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6
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Consider the matrix

 bold italic M equals open parentheses table row a 0 row c cell negative a end cell end table close parentheses 

where a comma space c element of straight real numbers . Find expressions for bold italic M to the power of 2 k end exponent and bold italic M to the power of 2 k plus 1 end exponent where k is a positive integer.

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7a
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Consider the matrix

 bold italic A equals open parentheses table row i 0 row 3 i end table close parentheses 

where i equals square root of negative 1 end root.

(i)
Find bold italic A squared comma space bold italic A cubed comma space bold italic A to the power of 4 and bold italic A to the power of 5.

(ii)
Hence determine bold italic A to the power of 15.

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7b
Sme Calculator3 marks

Find the general term for bold italic A double apostrophewhere n is a positive integer.

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8
Sme Calculator7 marks

Consider the 2 cross times 2 matrix

 bold italic M equals open parentheses table row a b row c d end table close parentheses 

Use algebra to find the requirements that must be satisfied by a comma space b comma space c and d in order for bold italic M squared equals open parentheses table row cell a squared end cell cell b squared end cell row cell c squared end cell cell d squared end cell end table close parentheses to be true.

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9a
Sme Calculator5 marks

A professional Football team are looking to buy new players.  Their scouts have returned a shortlist containing 23 English, 17 German, 18 Spanish, and 8 Italian players.

The shortlisted players are in the following proportions for each playing position: 

  • 11% are goalkeepers
  • 29% are wingers
  • 39% are defenders
  • 21% are strikers 

A given player only plays in one of the listed positions. 

(i)
Write a column matrix, N, representing the numbers of players from each country, and a row matrix, P, containing the proportions of players in each position.

(ii)
Hence find the total number of players in each position on the shortlist.

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9b
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Explain why it would be incorrect in general to say that the elements of the matrix NP  represent the numbers of players in each position by nationality. For example to say that (NP)1,1 (the entry in the first row and first column of matrix NP ) might represent the number of English goalkeepers.

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10a
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A ball is thrown vertically downwards from the top of a cliff, and its position is tracked from when it is first thrown until it hits the ground (at which point it may be assumed that the ball comes instantaneously to rest).

The height of the ball above the ground after t seconds is modelled by the equation s open parentheses t close parentheses equals a t squared plus b t plus c,  where a, b and c are real constants and the height s is measured in metres. After 1 second, the ball is 175.4 m above the ground; after 5 seconds, its height is 105 m; and after 6 seconds it is 74.4 m. 

Set up and solve a matrix equation to find

(i)
the height of the cliff
(ii)

the time taken for the ball to reach the ground.

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10b
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Another experiment studies the motion of another object that only moves in one dimension.  A quartic equation of the form s equals a t to the power of 4 plus b t cubed plus c t squared plus d t plus e is used to model the displacement of the object, where a comma space b comma space c comma space d and e are all real constants with a not equal to 0. 

(i)
State the number of measurements of the object’s displacement at different times that would need to be taken, in order to find the explicit values of the constants a comma space b comma space c comma space d and e.

(ii)
State the dimensions of the matrices involved in forming a matrix equation, as in part (a), to determine the constants for the quartic model.

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