Consider the following matrices:
where are constants.
Given that , find the values of x, y, z, and q.
Given that find the values of a, b, r, and s.
Given that find the values of c, d, e, and f.
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Consider the following matrices:
where are constants.
Given that , find the values of x, y, z, and q.
Given that find the values of a, b, r, and s.
Given that find the values of c, d, e, and f.
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Consider the two matrices
where is a constant.
Given that and are commutative, find the value of .
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Consider the matrix
Find an expression for .
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Consider the matrices:
where are constants.
Find the following products in terms of the appropriate constants. If it is not possible to do so, explain why.
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Consider the following matrices:
Show that and state the name of this property.
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For each of the following matrices,
(i) find the values of for which does not exist, and
(ii) for the cases where does exist, find in terms of .
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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 matrix. For example, “encode” becomes or . Then the message is multiplied, on the left, by an encryption matrix.
Here is a message which has been encoded using the encryption matrix :
Decode the message.
A spy (whose name cannot be given) notices that if the message has an odd number of letters, it will not completely fill a matrix. The spy suggests putting the message into a 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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Using the properties of matrices, explain why the following misconceptions are incorrect.
“I know that , therefore must also be true if and are matrices.”
“I know that , therefore must also be true if and are matrices.”
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In this question and are matrices, are constants, and is a positive integer.
A student claims it is always true that .
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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Consider the following matrices:
where are constants,
Given that and are commutative, find in terms of
Given that the determinant of is 26, find and .
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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.
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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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.
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 and represent the salaries for a veterinarian, a veterinary nurse, and an animal care assistant respectively:
= Total salary spend in thousands
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Consider a curve with equation , where a, b, and c are real constants. The graph passes through the points and
Consider a second curve with equation , where a, b, c, d, e and f are real constants with
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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Consider the following matrices:
where are constants.
Find each of the following matrix sums or differences, or if that is not possible then explain why:
Find each of the following matrix products, or if that is not possible then explain why:
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.
Find each of the following:
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Consider the matrices:
where is a constant.
Let be the identity matrix, and let 0 be the zero matrix.
Find the following
Find the following:
and are matrices such that and
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Consider the matrix
where is a constant.
Given that , find the value of .
Consider the two matrices
where are constants.
Given that , find the values of and .
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Consider the matrices and .
Find
Consider the matrix product .
The transition matrix of a dynamic system is .
Find .
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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
Burger store B sells their veggie burger, beef burger and chicken burger for $4.75, $5.85 and $5.50, respectively.
By first calculating the matrix , compare the total cost of the family’s dinner at stores A and B.
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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, , to represent the number of applications they received from advertising the job on social media and from using the hiring agency.
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, , to represent the percentages of the applicants that applied for each of the different jobs.
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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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 |
The price Amanda pays for each item is shown in the table below.
|
Tables |
Chairs |
Sofas |
Price |
$72.00 |
$14.50 |
$47.50 |
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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 and represent the amount paid by a student, an adult, and a pensioner respectively.
Write down a system of three linear equations in terms of and that represent the information shown in the table above.
Find the values of and using appropriate matrices and matrix inverses.
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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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.
Write down a system of linear equations that could be used to find the answers in part (a).
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The graph of the quadratic function passes through the points and .
Show that , and must satisfy the following system of linear equations:
Represent the system of equations in part (a) in matrix form.
Hence use a matrix method to find the values of , and .
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The graph of the function passes through the points and .
Write down a system of linear equations that and must satisfy.
Hence use a matrix method to determine the values of and .
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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 and represent the prices, in $/kg, for wheat, soybeans and sugar respectively.
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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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.
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 %, then the return is expected to be % 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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Consider the following matrices:
The matrix product 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.
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.
It is given that
Find the values of a, b, and t.
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In this question A, B, and C are arbitrary square matrices.
Prove the following matrix results, stating any necessary assumptions:
Using the result from part (a)(ii), simplify .
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Consider the matrices:
where are constants.
Given that M and N are commutative, find an expression for N in terms of b and d only.
Given that the inverse matrix exists
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Given that
find matrices B and C.
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For any matrices M, A or B:
Prove that , where is a real constant.
Prove that .
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Consider the matrix
where . Find expressions for and where k is a positive integer.
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Consider the matrix
where .
Find the general term for where is a positive integer.
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Consider the matrix
Use algebra to find the requirements that must be satisfied by and in order for to be true.
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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:
A given player only plays in one of the listed positions.
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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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 seconds is modelled by the equation , 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
the time taken for the ball to reach the ground.
Another experiment studies the motion of another object that only moves in one dimension. A quartic equation of the form is used to model the displacement of the object, where and are all real constants with
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