General Mathematics 2018 Chapter 5 - Matrices

Transcription

1 General Mathematics 2018 Chapter 5 - Matrices Key knowledge The concept of a matrix and its use to store, display and manipulate information. Types of matrices (row, column, square, zero, identity) and the order of a matrix. Matrix arithmetic: the definition of addition, subtraction, multiplication by a scalar, multiplication, the power of a square matrix, and the conditions for their use. Determinant and inverse of a matrix. Key skills Use matrices to store and display information that can be presented in rows and columns. Identify row, column, square, zero, and identity matrices and determine their order. Add and subtract matrices, multiply a matrix by a scalar or another matrix, raise a matrix to a power and determine its inverse, using technology as applicable. Use matrix sums, difference, products, powers and inverses to model and solve practical problems. Use of CAS calculator to do matrix operations Chapter Sections 5A The basics of a matrix 5B Using matrices to model(represent practical situations 5C Adding and subtracting matrices 5D Scalar multiplications 5E Matrix multiplication 5F Applications of matrices 5G Communications and connections 5H Identity and inverse matrices 5J Solving simultaneous equations 5K Extended application and problem solving tasks Transition Matrix Review Questions to be completed 1(a,b,c,d), 2(a,b,c), 3 and4(a,b) All questions All questions All questions All questions All questions All questions All questions All question All questions All questions All questions Page 1 of 28

2 Table of Contents 5A THE BASICS OF A MATRIX 3 ORDER OF A MATRIX 3 ELEMENTS OF A MATRIX 3 ROW MATRICES 4 COLUMN MATRICES 4 5B USING MATRICES TO MODEL/REPRESENT PRACTICAL SITUATIONS 6 Example 2: Using a matrix to represent connections. 6 5C ADDING AND SUBTRACTING MATRICES 7 ADDING MATRICES 7 SUBTRACTING MATRICES 7 Example 3: 7 THE ZERO MATRIX, 0 7 5D SCALAR MULTIPLICATION 8 Example 4: 8 Example 5: Application of scalar multiplication. 8 Example 6: Scalar multiplication and subtraction of matrices. 9 5E MATRIX MULTIPLICATION 10 RULES FOR MATRIX MULTIPLICATION 11 Example 7: 11 5F APPLICATIONS OF MATRICES 13 Example 8: Business application of matrices 13 PROPERTIES OF ROW AND COLUMN MATRICES 14 Example 9: Using row and column matrices to extract information 14 5G COMMUNICATION AND CONNECTIONS 15 Example 10: 15 Example: 17 5H IDENTITY AND INVERSE MATRICES 18 IDENTITY MATRIX 18 Example IDENTITY MATRIX 2 X 2 MATRICES 18 THE INVERSE OF A MATRIX AND ITS EVALUATION 19 ALTERNATIVELY, WE CAN USE CAS CALCULATOR TO CALCULATE THE DETERMINANT AND THE INVERSE OF A MATRIX 19 SINGULAR MATRIX 20 Example: 20 USING INVERSE MATRICES TO SOLVE PROBLEMS 20 5J SOLVING SIMULTANEOUS EQUATIONS USING MATRICES 21 Example 15: 21 Using the CAS 21 TRANSITION MATRICES APPLICATIONS 22 SETTING UP A TRANSITION MATRIX 22 Example16: 23 USING RECURSION TO GENERATE STATE MATRICES STEP-BY-STEP 23 CONSTRUCTING A MATRIX RECURRENCE RELATION 23 GENERATING S 1 24 GENERATING S 2 24 GENERATING S 3 24 Example 17: Using a recursion relation to calculate state matrices step-by-step 25 Example 18: 26 TRANSITION MATRICES QUESTIONS 27 Page 2 of 28

3 5A The basics of a matrix Matrices can be used to store information, solve sets of simultaneous equations, find optimal solutions in business, analyse networks, transform shapes in geometry, encode information and devise the best strategies in game theory. A matrix (plural matrices) is an array of numbers set out in rows and columns. Order of a matrix Matrices are described by the number of rows and number of columns. This is known as the order/size/dimension of a matrix. The above matrix has 2 rows and 3 columns and is called a 2 3 matrix. When writing a matrix, the number of rows is always given first followed by the number of columns. Matrices are usually named using capital letters such as A, B, X and Y. Elements of a matrix The number within a matrix are called its elements. For example, in the matrix: Element a13 is in row 1, column 3 and its value is 4 Element a22 is in row 2, column 2 and its value is 7 Page 3 of 28

4 Example 1: Interpreting the elements of a matrix. Matrix B shows the number of boys and girls in years 10 to 12 at a particular school. (a) Give the order of the matrix B. (b) What information is given by the element b12? (c) Which element give the number of girls in year 12? (d) How many boys in total? (e) How many students in year 11? Row Matrices A row matrix has a single row of elements. In matrix A, the Friday sales from the market stall can be represented by a 1 x 3 row matrix. Column Matrices A column matrix has a single column of elements. In matrix A, the sales of jeans from the market stall can be represented by a 2 x 1 column matrix. Page 4 of 28

5 Square Matrices In square matrices, the number of rows equals the number of columns. Here are three examples. Page 5 of 28

6 5B Using matrices to model/represent practical situations A network diagram consists of vertices and edges can be used to show connections or relationships between vertices. The information in network diagrams can be recorded in a matrix known as an adjacency matrix. Example 2: Using a matrix to represent connections. The diagram below shows the number of roads connecting between four towns, A, B, C and D. Construct an adjacency matrix to represent this information. TO A B C D F A R B A = O M C D Page 6 of 28

7 5C Adding and subtracting matrices Matrices can be added and subtracted if they have the same order/size/dimension. That is if they have the same number of rows and columns. Adding matrices To add matrices, add the corresponding elements of each matrix together. That is the numbers in the same position. Subtracting matrices To subtract matrices, subtract the corresponding elements of each matrix together. That is the numbers in the same position. Example 3: Complete the following addition and subtraction of matrices if possible. (a) [ ] + [ ] = (b) [ ] [ 1 9] = (c) [ ] + [ 2 8] = 1 0 The Zero Matrix, 0 In a zero matrix, every element is zero. The following are examples of zero matrices. Just like arithmetic with ordinary numbers, adding or subtracting a zero matrix does not make any changes to the original matrix. Also, subtracting any matrix from itself gives a zero matrix. Page 7 of 28

8 5D Scalar multiplication A scalar is just a number. Multiplying a matrix by a number is called scalar multiplication. Example 4: If A = [ ], find 3A 3A = 3 [ ] = Scalar multiplication has many practical applications. It is particularly useful in scaling up the elements of a matrix, for example, add the GST to the cost of the prices of all items in a shop by 10%, simply multiplying a matrix of prices by 1.1. (Note 1.1 is 110% where 100% which is the original price) plus 10% GST). Example 5: Application of scalar multiplication. A gymnasium has the enrolments in courses shown in this matrix. The manager wishes to double the enrolments in each course. Show this in a matrix. Page 8 of 28

9 Example 6: Scalar multiplication and subtraction of matrices. Calculate 2 [ ] 3 [ ] Page 9 of 28

10 5E Matrix Multiplication Matrix multiplication is the multiplication of a matrix by another matrix. The matrix multiplication of two matrices A and B can be written as A B or just AB. Although it is called multiplication and the symbol may be used, matrix multiplication is not as simple as multiplication of numbers but a routine involving the sum of pairs of numbers that have been multiplied. For example, the method of matrix multiplication can be demonstrated by using a practical example. The numbers of CD s and DVD s sold by Fatima and Gaia are recorded in matrix N. The selling prices of the CD s and DVD s are shown in matrix P. We want to make a matrix, S, that shows the value of the sales made by each person. The steps used in this example follow the routine for the matrix multiplication of N P Page 10 of 28

11 Rules for matrix multiplication Because of the ways the products are formed, the number of columns in the first matrix must equal the numbers of rows in the second matrix. Otherwise, we say that matrix multiplication is not defined, meaning it is not possible. In our example of the CD and DVD sales: Notice that the outside numbers give the order of the product matrix (answered matrix). It is made by multiplying the two matrices. In our case, the order of the answer matrix is 2 x 1. Example 7: 5 2 For the following matrices A = [ 4 6] B = [ ] C = [ 2 4 7] D = [ 6] (i) (ii) (a) AB decide whether the matrix multiplication in each question below is defined. if matrix multiplication is defined, give the order of the answer matrix and then do the matrix multiplication. Multiplication of matrix test A B AB Matrix order (b) BA Multiplication of matrix test A B AB Matrix order Page 11 of 28

12 (c) CD Multiplication of matrix test M N MN Matrix order Usually, when we reverse the order of the matrices in matrix multiplication, we get a different answer. That is A B does NOT equal to B A. Define matrices A and B On a calculator page, press: MENU b 1: Actions 1 1: Define 1 The template for the (2 2) can be found by pressing / t Complete the entry lines as: Define M = [ ] Define N = [ ] Press ENTER after each entry Complete the entry lines as: M N N M Press ENTER after each entry. Page 12 of 28

13 5F Applications of matrices Data represented in matrix form can be multiplied to produce new useful information. Example 8: Business application of matrices Fatima and Gaia s store has a special sales promotion. One free cinema ticket is given with each DVD purchased. Two cinema tickets are given with the purchase of each computer game. The number of DVDs and games sold by Fatima and Gaia are given in matrix S. The selling price of a DVD and a game, together with the number of free tickets is given by matrix P. Page 13 of 28

14 Properties of row and column matrices Row and column matrices provide efficient ways of extracting information from data stored in large matrices. Matrices of a convenient size will be used to explore some of the surprising and useful properties of row and column matrices. Example 9: Using row and column matrices to extract information Three rangers completed their monthly park surveys of feral animal sightings in the matrix S. (a) Evaluate S B (b) What information about matrix S is given in the product S B? (c) Evaluate A S (d) What information about matrix S is given in the product A S? Page 14 of 28

15 5G Communication and connections Social networks, communication pathways and connections can be represented and analysed using matrix techniques. Example 10: The diagram shows the communications within a group of friends, where: a double-headed arrow connecting two names indicates that those two people communicate with each other. if there is no arrow directly connecting two people, they do not communicate. (a) These links are called one-step connections because there is just one direct step in making contact with the other person. Record the social links in a matrix N, using the first letter of each name to label the columns and rows. Explain how the matrix should be read. TO V S K P F V R S N = O M K P (b) Explain why there is a symmetry about the leading diagonal of the matrix. Page 15 of 28

16 (c) What information is given by the sum of a column or row? (d) N 2 gives the number of two-step communications between people. Namely, how many ways one person can communicate with someone via another person. Find the matrix N 2, the square of matrix N. N = N 2 = (e) Use the matrix N 2 to find the number of two-step ways Kathy can communicate with Steven and write the connections. (f) In the N 2 matrix there is a 3 where S column meets the S row. This indicates that there are three two-step communications Steven can have with himself. Explain how this can be given a sensible interpretation. The matrix N 3 would give the three-step communications between people. The number of ways of communicating with someone via two people. The matrix methods of investigating communications can be applied to friendships, travel between towns and other types of two-way connections. Page 16 of 28

17 Example: The following adjacency matrix shows the number of pathways between four attractions at the zoo: Lions (L), Seals (S), Monkeys (M) and Elephants (E). Using CAS or otherwise, determine how many ways a family can travel from the Lions to the Monkeys via one of the other two attractions. Solution The number of ways a family can travel from the Lions to the Monkeys via one of the other two attractions indicates that we need to determine a path matrix. A simple way of determining a two-steps path matrix is simply raise the matrix to the power of. A 2 = = 2 Reading down the L column to the M row in the two steps matrix, A 2, we can say that there are ways in which a family can travel from the Lions to the Monkeys via one of the other two attractions. Page 17 of 28

18 5H Identity and Inverse Matrices Identity matrix In ordinary arithmetic, when 1 multiplies a number, the answer is always the original number. Example 3 1 = 3, the number 1 is called the multiplicative identity element. When multiplying any matrix by an identity matrix the resultant matrix will be the original matrix. Example 11 Given two matrices A = [ ] and I = [ ] (a) Calculate AI = [ ]. [ ] = (b) Calculate IA = [ ]. [ ] = Identity matrix 2 x 2 matrices The identity matrix, I, also has the special property that it is commutative in matrix multiplication. When I is one of the matrices in the multiplication, the answer is the same when the order of the matrices is commuted (reversed). So AI = IA = A Remember that matrix multiplication is not usually commutative, except for this special case Only a square matrices have identity matrices. The identity matrix for any square matrix is a square matrix of the same order with 1s along the leading diagonal (from the top left to the bottom right) and 0s in all other positions. Page 18 of 28

19 The inverse of a matrix and its evaluation In the real number system, a number multiplied by its reciprocal results in 1, example = 1. In this case 1 is the reciprocal or multiplicative inverse of 3. 3 In matrices, if the product matrix is the identity matrix, then one of the matrices is the multiplicative inverse of the other. [ ]. [ ] = [ ] = I A A 1 = I = A 1 A The inverse matrix A, written as A 1, is a matrix that multiplies A to make the identity matrix I Finding the inverse matrices of the matrix A = [ ] Step 1: Determine the determinant of matrix A Det A = A = (8 2) (4 3) = = 4 Step 2: Swap the elements in the main the diagonal [ 2 8 ] Step 3: Multiply the elements on the other diagonal by 1 or simply swap the signs of these numbers Step 4: Write the inverse matrix of A [ ] Determinant A 1 = 1 [ ] = [ ] = [ ] = [ ] Only square matrices have inverses Alternatively, we can use CAS calculator to calculate the determinant and the inverse of a matrix Define matrices A On a calculator page, enter as follows to determine the determinant of matrix A det(a) or det( ) Define matrices A On a calculator page, enter as follows to determine the determinant the inverse of matrix A a 1 or [ ] 1 Page 19 of 28

20 Singular Matrix Example: Determine the inverse for the matrix B = [ ] Calculate the determinant of B = Calculate the inverse of B 1 = Since the determinant of matrix B is equal to, therefore the inverse of B does not. Therefore, matrix B is said to be matrix. Using inverse matrices to solve problems Unlike in the real number system, we can t divide one matrix by another matrix. However, we can use inverse matrices to help us solve matrix equations in the same way that division is used to help solve many linear equations. Given the matrix equation AX = B, where matrix X is the unknown. We can use the multiplication of the inverse matrix to find the value of the unknown matrix X as follows: If the equation was XA = B, where the matrix X is the unknown that is needed to be found. Again we can apply the principal of inverse multiplication to determine for the value of matrix X: Page 20 of 28

21 5J Solving Simultaneous equations using matrices If you have a pair of simultaneous equations, they can be set up as a matrix equation and solved using inverse matrices. Example 15: Solve the following pair of simultaneous equations by using inverse matrices. 2x + 3y = 6 4x 6y = 4 Set up the simultaneous equations as a matrix equation then solve for the value of the unknowns Using the CAS Open a Calculator page and complete the entry lines as: [ ] a [ 6 4 ] b Press ENTER after each entry. X is found by pre-multiplying both sides of the equation by A -1 (and hence isolating X on the left and leaving A -1 B on the right). Complete the entry line as: a -1 b Then press ENTER Alternatively, you can use the solve function on the CAS Type the following: solve ([ ] [x y ] = [ 6 4 ], x) Then press ENTER Page 21 of 28

22 Transition Matrices Applications Setting up a transition matrix A car rental firm has two branches: one in Bendigo and one in Colac. Cars are usually rented and returned in the same town. However, a small percentage of cars rented in Bendigo each week are returned in Colac, and vice versa. The diagram below describes what happens on a weekly basis. What does this diagram tell us? From week to week: 0.8 (or 80%) of cars rented each week in Bendigo are returned to Bendigo 0.2 (or 20%) of cars rented each week in Bendigo are returned to Colac 0.1 (or 10%) of cars rented each week in Colac are returned to Bendigo 0.9 (or 90%) of cars rented each week in Colac are returned to Colac The percentages (written as proportions) are summarised in the form of the matrix below. This matrix is an example of a transition matrix (T). It describes the way in which transitions are made between two states: state1: the rental car is based in Bendigo. state2: the rental car is based in Colac. Note: In this situation, where the total number of cars remains constant, the columns in a transitional matrix will always add to one (100%). For example, if 80% of cars are returned to Bendigo, then 20% must be returned to Colac. Page 22 of 28

23 Example16: A factory has many different machines. The machines can be in one of two states: operating or broken. Broken machines are repaired and come back into operation, and vice versa. On a given day: 85% of machines that are operational stay operating 15% of machines that are operating breakdown 5% of machines that are broken are repaired and start operating again 95% of machines that are broken stay broken. Construct a transition matrix to describe this situation. Use the columns to define the situation at the Start of the day and the rows to describe the situation at the End of the day. Transition Diagram Transition matrix Using recursion to generate state matrices step-by-step We return to the car rental problem discussed above. The car rental firm now plans to buy 90 new cars. Fifty will be based in Bendigo and 40 in Colac. Given this pattern of rental car returns, the first question the manager would like answered is: If we start with 50 cars in Bendigo, and 40 cars in Colac, how many cars will be available for rent at both towns after 1 week, 2 weeks, etc? If you think of a number and double it, over and over again, you are just taking the current number and timing it by We do the same with the car rental problem, the only difference being that we are now working with matrices. Constructing a matrix recurrence relation A recurrence relation must have a starting point. In this case, it is the initial state matrix: S 0 = [ ] Bendigo Colac Page 23 of 28

24 Generating S 1 To find out the number of cars in Bendigo and Colac after 1 week, we use the transition matrix T = [ ] to generate the next state matrix in the sequence, S, as follows: S 1 = TS = [ ] [50 40 ] = [ ] S 1 = [ ] Bendigo Colac Thus, after 1 week we predict that there will be 44 cars in Bendigo and 46 in Colac. Generating S 2 Following the same pattern, after 2 weeks; S 2 = TS 1 = [ ] [44 46 ] = [ ] = [40 50 ] Bendigo Colac Thus, after 2 weeks we predict that there will be 40 cars in Bendigo and 50 in Colac. Generating S 3 After 3 weeks: S 3 = TS 3 = [ ] [ ] = [ ] = [37 53 ] Bendigo Colac Thus, after 3 weeks we predict that there will be 37 cars in Bendigo and 53 in Colac. A pattern is now emerging. So far, we have seen that: S 1 =TS 0 S 2 =TS 1 S 3 =TS 2 If we continue this pattern we have: S 4 =TS 3 S 5 =TS 4 or, more generally, S n+1 = TS n. With this rule as a starting point, we now have a recurrence relation that will enable us to model and analyse the car rental problem on a step-by-step basis. Page 24 of 28

25 Example 17: Using a recursion relation to calculate state matrices step-by-step A factory has a large number of machines. The machines can be in one of two states: operating (O) or broken (B). Broken machines are repaired and come back into operation and vice versa. At the start, 80 machines are operating and 20 are broken. Use the recursion relation S 0 = initial value, S n+1 = T S n Where S 0 = [ ] and T = [ ] to determine the number of operational and broken machines after 1 day and after 3 days Calculator hint: In practice, generating matrices recursively is performed on your CAS calculator as shown opposite for the calculations performed in last the example. Page 25 of 28

26 Example 18: In a large country town, there are three major supermarkets. Customers switch from one to another due to advertising, better service, prices and for other reasons. A survey of 1000 customers has revealed the following information for the past month. Best buys started with 40% of the market; 90% of its customers remained loyal to Best Buys but 5% changed to Great Groceries and 5% to Super Store. Great Groceries started with a 36% market share: 85% remained loyal, 10% transferred to Best Buys and 5% to Super Store. Super Store stared with 24% of the customers: it lost 15% to Best Buys and 5% to Great Groceries, but 80% remained. Summarise the information in matrix form and calculate the new market share. Page 26 of 28

27 Transition Matrices Questions Question 1 Two politicians, Rob and Anna, are the only candidates for a forthcoming election. At the beginning of the election campaign, people were asked for whom they planned to vote. The numbers were as per the table. During the election campaign, it is expected that people may change the candidate that they plan to vote for each week according to the transition diagram shown. (a) The total number of people who are expected to change the candidate that they plan to vote for 1 week after the election campaign begins is: A. 828 B C D E (b) The election campaign will run for 10 weeks. If people continue to follow this pattern of changing the candidate they plan to vote for, the expected winner after 10 weeks will be: A. Rob by about 50 votes B. Rob by about 100 votes C. Rob by fewer than 10 votes D. Anna by about 100 votes E. Rob by about 200 votes Question 2 At a large retail outlet, 60% of people drink coffee and 40% drink tea. The catering company has decided to introduce a new brand of coffee and market research shows that of those who drink tea 45% will change to coffee each week and of those who drink coffee only 10% will change to tea each week. The remainder will continue to drink the same drink as present. (a) Draw a tree diagram to represent this situation for week 1. (b) What proportion of people will drink coffee at the end of week 1? (c) Set up matrices to represent the situation. (d) Solve the matrices to show that you get the same answer as in part (b). (e) What proportion of people will drink coffee at the end of week 3? Page 27 of 28

28 Question 3 At a large retail outlet, 55% of people drink coffee and 45% drink tea. The catering company has introduced a new brand of tea and market research shows that of those who drink tea 15% will change to coffee each week and of those who drink coffee 75% will change to tea each week. (a) Draw a tree diagram to represent this situation for 1 week. (b) What proportion of people will drink coffee at the end of 1 week? (c) Set up matrices to represent this situation. (d) Solve the matrices to show that you get the same answer as in part (b). Question 4 At a large retail outlet, only two types of milkshake are produced. At the moment, 45% of people drink chocolate milkshakes and 55% drink strawberry milkshakes. The catering company has decided to introduce a richer strawberry milkshake in place of the current one, and market research shows that of those who drink chocolate milkshakes 35% will change to strawberry each month and of those who drink strawberry only 5% will change to chocolate each month. (a) What is the initial state matrix? (b) What is the transition matrix? (c) What proportion of people will drink each type of milkshake at the end of 1 month? (d) What proportion of people will drink each type of milkshake at the end of 2 months? (e) What proportion of people will drink each type of milkshake at the end of 3 months? (f) What proportion of people will drink each type of milkshake at the end of 100 months? (g) What proportion of people will drink each type of milkshake at the end of 101 months? (h) What do you notice about the answers for (f) and (g)? Question 5 24% of students in a large school own a Warren mobile phone and the rest own an Oval mobile phone. The company that owns Warren decided to run a series of advertisements to promote Warren and market research shows that 15% of students who own an Oval mobile phone will change to Warren each month and 10% of students who own a Warren mobile phone will change to Oval each month. Assume they are all on monthly plans. (a) What is the initial state matrix? (b) What is the transition matrix? (c) What proportion of the students will use each type of phone at the end of 1 month? (d) What proportion of the students will use each type of phone at the end of 2 months? (e) What proportion of the students will use each type of phone at the end of 3 months? (f) What proportion of the students will use each type of phone at the end of 50 months? (2dp) (g) What proportion of the students will use each type of phone at the end of 51 months? (2dp) (h) What do you notice about the answers for (f) and (g)? Question 6 35% of students travel by train to a certain school and the rest travel by bus. Vic Rail decided to offer a huge discount to school students to increase their market share. It is known that 20% of students who travel by bus will switch to travelling by train and only 5% of students who travel by train will switch to travelling by bus. (a) What is the initial state matrix? (b) What is the transition matrix? (c) What proportion of the students will use each type of transport at the end of 1 month? (d) What proportion of the students will use each type of transport at the end of 2 months? (e) What proportion of the students will use each type of transport at the end of 3 months? (f) What proportion of the students will use each type of transport at the end of 3 years? Give your answer correct to 2 decimal places. Page 28 of 28