Unit Five Matrices and Networks 15 HOURS MATH 521A

Transcription

1 Unit Five Matrices and Networks 15 HOURS MATH 521A Revised Mar 16,

2 SCO: By the end of grade 11 students will be expected to: A13 demonstrate an understanding of the properties of matrices and apply them Elaborations - Instructional Strategies/Suggestions Matrices Initiate a discussion on the concept of matrices. Students should become familiar with such terms as: elements - individual numbers in a matrix rows - horizontal groups of numbers in a matrix columns - vertical groups of numbers in a matrix dimensions - number of rows and columns in a matrix naming a matrix A matrix is a rectangular array of numbers within brackets. The array is used to represent real world data and solve real world problems. Simply put, it is a way to arrange data in a table form. The discussion could also touch on types of matrices such as: square, row, column, zero and identity. Invite students to explore the matrix feature on the TI-83. Example: enter the matrix into the TI-83 matrix < edit enter the dimensions 2 3 then enter the elements in matrix A. 108

3 Matrices Pencil/Paper A store sells two types of sneakers, cross-trainers and court sneakers. In June, the store sold 50 cross-trainers and 30 court sneakers, while in July they sold 80 cross-trainers and 90 court sneakers. Represent this information in a rectangular array(or matrix form). Matrices Student Workbook #1-7 Pencil/Paper A music store compared the sales of Rap music CD s to classical music over 3 months. In November, the store sold 70 Rap CD s and 100 classical CD s. In December, they sold 120 Rap CD s and 90 classical CD s. Finally in January there were 80 Rap CD s and 60 classical CD s sold. Represent this information in matrix form. Group Project/Communication Give three examples of situations where information can be displayed in matrix form. Write a matrix for each situation. Research Investigate the life of Arthur Cayley and write about his contributions to mathematics. 109

4 Matrices (cont d) Pencil/Paper In matrix Matrices (cont d) a) state the dimensions of matrix A b) what is the element in row 2 column 3 c) what is the element in row 3 column 4 Pencil/Paper A store sells three sneaker brands; Nike, Reebok and Adidas. In May there were 60 Nike, 30 Reebok and 40 Adidas pairs sold. In June there were 70 Nike, 80 Reebok and 25 Adidas pairs sold. Represent this information in matrix form (brand month). 110

5 Matrices (cont d) Research Bring to class 2 examples of matrices that you have found in the newspaper or a magazine. Only one of the examples can be sports-related. It is difficult to read through a newspaper and not see examples of matrices. Below are just a few examples. Any table that has rows and columns is a matrix. Databases are examples of matrices used to organize information in matrix form. 111

6 SCO: By the end of grade 11 students will be expected to: A13 demonstrate an understanding of the properties of matrices and apply them Elaborations - Instructional Strategies/Suggestions Matrix Addition Allow groups to explore using the TI-83 to discover the rules as to when matrices can be added or subtracted. Students should be able to induce that only elements in matching positions in each matrix can be added and therefore the matrices must have the same dimensions for them to be added or subtracted. Ex. 1 in Worthwhile Tasks Enter the matrices A and B into the TI-83 as shown on p.2 Once the matrices have been entered, press 2 nd quit to return to the home screen. To add the matrices: Press matrix 1:A press enter + matrix? down to 2:B press enter Press enter Note: The same procedure can be followed if matrices are to be multiplied. 112

7 Matrix addition Pencil/Paper/Technology Use the following problems to complete the table below, if possible. Once the table is completed, look for a pattern and with inductive reasoning state a rule for matrix addition or subtraction. 1) Matrix addition Student Workbook #8 Algebra, Structure and Method Book2 p.773 #1-8 Find A + B 2) Find A+B 3) Find A - B 4) Find A + B 113

8 SCO: By the end of grade 11 students will be expected to: A13 demonstrate an understanding of the properties of matrices and apply them A14 explain why and apply the fact that matrices are not commutative under multiplication Elaborations - Instructional Strategies/Suggestions Multiplication (scalar and matrix) 1) Scalar Allow students to investigate multiplication of a matrix by a scalar (a real number). Allow students to brainstorm ways of solving this type of problem. One method students may suggest is using the TI-83. 2)Matrix Invite student groups to complete the table in the Worthwhile Tasks and formulate a rule for matrix multiplication. Note to Teachers: Multiply: To see if these matrices can be multiplied get the dimensions of each: B5 B6 develop, analyze and apply procedures for matrix multiplication solve network problems using matrices 1 st matrix 2 nd matrix row column row column if the 2 inside numbers (means) are the same then the matrices can be multiplied yielding a matrix with dimensions determined by the outer numbers above (extremes). For the above example the inner numbers are both 2 and thus multiplication can be done. The outer numbers are 1 and 2 and thus the dimensions of the solution matrix is 1 2 (1 row and 2 columns). B9 use the calculator correctly and efficiently for various computations C8 represent network problems using matrices To fill in these blanks name their positions. The first blank is in the 1 st row, 1 st column position. To get the element that goes in this blank multiply the elements in the first row of matrix A by the elements in the first column of matrix B. ie: (1!4) + (5 5) = 21 Thus the first blank is the element 21 The second blank has the position, 1 st row 2 nd column. The element that goes here comes from multiplying the elements in the 1 st row of matrix A and the 2 nd column of matrix B. ie: (1 2) + (5 1) = 7 Thus the second blank is the element 7 114

9 Matrix Multiplication 1) Scalar Pencil/Paper A store sells two types of sneakers, cross-trainers and court sneakers. In June, the store sold 50 cross-trainers and 30 court sneakers, while in July they sold 80 cross-trainers and 90 court sneakers. Represent this information in a rectangular array(or matrix form). If sales were twice the original projections, represent this solution in matrix form. Matrix Multiplication 1)Scalar Student Workbook #9,10 Algebra, Structure and Method B Book 2 p.773 # 9,10,13,15,16 Journal Does scalar multiplication change the dimensions of a matrix? 115

10 2) Matrix multiplication Technology Use the following problems to complete the accompanying table, if possible. Look for a pattern and induce a rule for matrix multiplication. When can it be done and how can it be done. Find the solution for A B : 1) 2) Matrix multiplication Matrix Student Workbook #11-18 p.777 oral # 1-10 written #1-10 Mathematical Modeling p ) 3) 4) 5) 6) 116

11 Matrix multiplication (cont d) 7) Technology/Communication In the above series of problems, find the solution matrix for B A, if possible. Are the answers the same as those for A B? 117

12 Matrix multiplication (cont d) Pencil/Paper/Technology (continuation of a problem from page 3) A store sells two types of sneakers, cross-trainers and court sneakers. In June, the store sold 50 cross-trainers and 30 court sneakers, while in July they sold 80 cross-trainers and 90 court sneakers. Represent this information in a rectangular array(or matrix form)(brand month). Now you know that the price of sneakers is Nike $90, Reebok $70 and Addidas $85. Write this in matrix form (price brand). Finally multiply these matrices to determine the revenue generated by each brand each month. Solution Activity Create a scenario where information can be written in 2 matrices and multiplication of the matrices presents a solution to the real world situation you have created. Explain what this third matrix represents. 118

13 Matrix multiplication (cont d) Pencil/Paper/Technology Two outlets of an electronics store sell 3 comparable items. Use matrix multiplication to show the total revenue that these items could generate in each store when they are sold at the regular price and at the sales price. Solution N = number of items P = prices of items Journal Explain the method you used to solve this problem. 119

14 Pencil/Paper/Technology Write the following information from the CFL in matrix form and label it matrix A.. W L T Toronto Montreal B. C Calgary a) What are the dimensions of this matrix? b) Generate matrix B representing the points awarded for a win, a loss and a tie. c) Calculate A B d) What do the elements in the product matrix represent? Pencil/Paper/Technology Your teacher keeps a record of your marks in matrix form with rows representing students and columns representing the test results in %. Class tests/assignments are worth 60% of the term mark while the final exam is worth the remaining 40%.In this example there are 5 class tests worth 60% 5 = 12% (.12) each. The final exam is worth 40% (.40). a) Enter the information below into matrix A in the TI-83 b) Create matrix B representing the values of the tests. Enter this into matrix B in the TI-83. c) Calculate A B. d) What do the elements in the product matrix represent? The number of points for each team. The elements in the product matrix represent the final marks for each student. 120

15 SCO: By the end of grade 11 students will be expected to: C7 C8 model real world situations with networks represent network problems using matrices E1 represent network problems as di-graphs Elaborations - Instructional Strategies/Suggestions Networks and Network di-graphs Allow students to discuss ideas about the term network. Student ideas could be written on the board and could include: a network of friends a computer network a network of roads the discussion could centre on what these ideas have in common. In all examples there are connections among things or people. Connectivity is the focus of this section. A network is a number of people, cities, computers, etc. that are connected in some way. Networks can be represented pictorially (graphically) in order to visualize some situation. The networks we will be looking at will be composed of closed loops. Network graphs consist of vertices and edges. Vertices represent the items (people, computers, towns, airports, etc.) In the network. Edges represent the connections (phone lines, roads, airline flight paths, etc.) Between the vertices. An even vertex is one where an even number of edges meet. An odd vertex is one where an odd number of edges meet. Note to Teachers: Murphy s Law states that if a network has odd vertices then it must have an even number of odd vertices. That means you cannot construct a network with one odd vertex (or 3 odd vertices, etc.) If a network does have odd vertices, then it must have only two odd vertices for the network to be completely covered traveling each edge only once, starting and ending at different odd vertices. Worthwhile Tasks for Instruction and/or Assessment Suggested Resources 121

16 Networks and Network di-graphs Pencil/Paper Each of the network graphs below represents roads between different communities. Complete the table by finding the number of roads that meet at each vertex(community). Number of roads that meet at each vertex Networks Student Workbook # ) 1) 2) Solutions 3) 4) Journal Which of the above networks have all vertices even? Which have 2 odd or 4 odd? Group Activity Construct a network with only one odd vertex. Note to Teachers: A network with only one odd vertex can t be drawn. 122

17 SCO: By the end of grade 11 students will be expected to: Elaborations - Instructional Strategies/Suggestions Networks and Network di-graphs (cont d) Challenge students to completely cover each network, covering each edge (road) exactly once. Vertices (towns) may be re-visited. Encourage a discussion to ensue on which networks this can be done when; a) starting and stopping at the same vertex (town) b) starting and stopping at different vertices. The students have done network problems where they had to determine the number of roads at each vertex and thus determine if the vertices are even or odd. Once they have tried to cover these networks without repeating any roads the above discussion should have students induce that: Networks that have all even vertices can be covered when starting and stopping at the same vertex. Problems (1) and (5) Networks that have 2 odd vertices can be covered when starting at one odd vertex and ending at the other odd vertex. Problems (2) and (4) Note: Have the students write the answer in two standard forms: a) list the vertices in order from the starting vertex to the ending vertex b) on the diagram, label the start and end vertices, and put arrows on the edges showing the path followed. Once arrows have been placed on the network, it has been turned into a di-graph (direction graph). Example: Can part of the network of roads below be traveled exactly once by: a) starting and stopping at the same vertex b) by starting and stopping at different vertices Show one possible trip for both (a) and (b). Each solution must contain at least 5 vertices and be a closed network. 123

18 Networks and Network di-graphs (cont d) Pencil/Paper Which of the above 5 networks on p.16 can be traveled without repeating a road by entering and exiting at different vertices? Do you notice a pattern? Solution In the problems on page 16, the networks that can be traveled without repeating a road by entering and exiting at the same vertex are numbers (1) and (5). Student Workbook #23-34 (1) one possible solution is: enter at A 6C 6D 6E 6C 6B 6A (5) one solution is: D 6C 6A 6B 6C 6A(circle) 6D Pencil/Paper Which of the above 5 networks can be traveled without repeating a road by entering and exiting at different vertices? Do you notice a pattern? Solution It can only be done when a network has 2 odd vertices. You must enter and exit at different odd vertices. (2) one solution is: B 6A 6D 6B 6C 6D (4) one solution is: B 6A 6F 6E 6D 6B 6C 6D Presentation/Communication Have groups create two more networks such that all edges can be covered exactly once; a) entering and exiting at the same vertex b) entering and exiting at different vertices Discussion Modify problem # 2 on p.16 so that you can cover all edges exactly once, starting and stopping at the same vertex. Presentation/Communication Have groups create two network di-graphs such that all edges can be covered exactly once; a) entering and exiting at the same vertex b) entering and exiting at different vertices 124

19 SCO: By the end of grade 11 students will be expected to: Elaborations - Instructional Strategies/Suggestions Network and Network di-graph Adjacency Matrices Invite students to discuss the idea of adjacent. Then allow students to brainstorm on what the term adjacency matrix might possibly mean. An adjacency matrix shows all possible connections between adjacent vertices in a network or network di-graph. (ie. The number of direct routes between vertices). Normally it is labeled as matrix A. Challenge student groups to form adjacency matrices for the problems in the Student Workbook. Example: Create the adjacency matrix for this network: solution: Create the adjacency matrix for this network di-graph of flights between Maritime cities: C = Charlottetown M = Moncton H = Halifax solution Invite the students to calculate A 2 and investigate the geometric meaning of it. A 2 is a matrix that represents the number routes from one vertex to another with one stop on the way ( a flight from Charlottetown to Toronto where a stop-over in Halifax is made, not a direct flight). This can be calculated easily using the TI-83. A 3 is a matrix where there are two stop-overs on the way from one vertex to another. 125

20 Pencil/Paper Create the adjacency matrix A for problem (1) on page 6. Solution (1) Student Workbook # A = Pencil/Paper The network graph below represents locations in a city connected by one way streets. This type of network graph is called a di-graph (direction graph). Create the adjacency matrix for this network di-graph. solution Journal Explain the term adjacency matrix. Pencil/Paper Create a di-graph for this adjacency matrix. solution Technology For the above problem, calculate the A 2 and A 3 matrices and explain their meaning. 126

21 SCO: By the end of grade 11 students will be expected to: A12 demonstrate an understanding of the conditions under which matrices have identities and inverses B9 use the calculator correctly and efficiently for various computations Elaborations - Instructional Strategies/Suggestions Identity Matrix Invite students to examine and discuss the identity element for multiplication. Students should be able to deduce that the identity element is 1 (ie. An expression or number multiplied by the identity element remains the same. In matrix theory, the identity element is actually a matrix called the Identity Matrix. Allow students to use the TI-83 to find the 2 2 and 3 3 Identity Matrices. Students should notice that these matrices must be square. Determinant of a Matrix Each square matrix has associated with it a real number called a determinant. It is usually displayed in the same format as a matrix but with vertical bars instead of brackets. The determinant will be used later in this unit to calculate the inverse of a matrix. Both of these will be applies to solving systems of linear equations at the end of the unit. For a 2 2 matrix: For a 3 3 matrix: To get the second line above: < pick a row or column (here we have picked the top row) < pick the top left element a and multiply it by the determinant of the correct 2 2 minor. That minor is found by crossing out the row and column that a is part of. < pick the next element in the top row b and change its sign (the sign must be changed for every second element in this row). Multiply it by the correct minor ( cross out the row and column containing b ) < pick element c and multiply it by its appropriate minor. This method can be used to evaluate the determinant for a square matrix of any dimension. 127

22 Identity matrix Performance Using Guess and Check attempt to generate the 2 2 and 3 3 Identity Matrices. Technology Use the TI-83 to determine the 2 2 and 3 3 Identity Matrices. Solution matrix < math 5: identity(2) Identity Matrix Student Workbook #44-46 This can be repeated for a 3 3 matrix. Pencil/Paper/Technology Create any 2 2 matrix and name it A. Explore the solutions to AA I and IAA. Write a few sentences explaining the answers you obtained. Determinant of a Matrix Pencil/Paper/Technology Evaluate each of the following: 1) 4) Determinant of a Matrix Student Workbook #47 Algebra, Structure and Method, Book 2 Using the TI-83 p.789 #1-17 p.800 #1-9 2) 5) 3) 6) matrix < math 1:det matrix 1:A) then press enter 7) 8) 9) 128

23 SCO: By the end of grade 11 students will be expected to: A12 demonstrate an understanding of the conditions under which matrices have identities and inverses Elaborations - Instructional Strategies/Suggestions Inverse of a Matrix Challenge student groups to brainstorm on their understanding of inverse. Although the idea of additive inverse may come up in the discussion, here we want to concentrate on the multiplicative inverse. Students should be able to come up with the idea that an expression multiplied by its inverse equals 1. Allow student groups to get the inverse of a matrix by Guess and Check. The formula for the inverse of a matrix is:,then A -1 = B17 derive, analyze and apply the procedure to obtain the inverse of a matrix Students should try using the TI-83 to get the inverse of 2 2 and 3 3 matrices or any square matrix. An example is: B9 use the calculator correctly and efficiently for various computations matrix 1:A x!1 enter 129

24 Inverse of a Matrix Pencil/Paper/Technology Find the inverse of each of the following. Then multiply these answers by the original matrix. Then use the TI-83 to confirm your result. Inverse of a Matrix Student Workbook #48-51 Alg, stru and Method Book 2 p.792 #1-6 oral ex. p.792#1-8 written ex. 1) 4) 2) 5) 3) 6) Jou rnal What pattern have you noticed in the above problems? What is its meaning? Presentation Without solving for the inverse, predict whether each matrix has an inverse. 1) 4) 2) 5) 3) 6) Journal Explain how, by quickly looking at a matrix, you can determine if that matrix has an inverse. 130

25 SCO: By the end of grade 11 students will be expected to: A15 apply inverses of a matrix when solving systems of equations Elaborations - Instructional Strategies/Suggestions Solving Systems of Linear Equations using matrices Algebraically A teacher might ask their class if anyone can explain how a system might be solved using matrices. This may lead to a discussion about the use of matrices in solving real world problems quickly and efficiently. For the teacher: Looking at a system like written in matrix form is: ax+by=e cx+dy=f B16 derive, analyze and apply matrix procedures to solve 2 2 systems with and without technology If we write it as < then multiply both sides by the inverse of A (A! 1 ) we get the solution for x and y A X = B A!1 A X = A!1 B X = B B17 derive, analyze and apply the procedure to obtain the inverse of a matrix B9 use the calculator correctly and efficiently for various computations For an explanation of the above see Algebra, Structure & Method Book 2 p.791 Ex. 1. x! 2y = 7 3x+ 4y = 1 matrix < edit enter put in the matrix dimensions and elements you have now matrix A filled in edit matrix B by pressing: matrix < edit? 2:B enter put in the dimensions and elements Press 2 nd quit and press the keystrokes below: matrix A x -1 matrix? 2:B enter So the solution is (3,!2) 131

26 Solving using matrices Pencil/Paper Algebraically Solve algebraically 2 ways: Solve using matrices: 1) 5x! 3y = 9 2) x + 5y =!11 4x! 3y = 25 2x! 5y =!4 Technology Solve using the TI-83 1) 2s + 3t = 6 2) 2x + y =!5 5s +10t=20 3x + 5y = 3 Applications Solve using matrices and by graphing on the TI -83: 1. A store owner can buy a second hand vending machine for $120. If a bottle of pop can be bought for $.65 and sold for $.95, how many bottles must be sold to recover the costs incurred? Note; this is called the break-even point (where expenses exactly equals income). It is used to help businesses estimate the amount of a product that must be sold before a profit can be made. Solution x = # bottles of pop sold y = dollar value (Cost) y = x (Revenue) y =.95x Method 5 - Solving using matrices Student Workbook # Algebraically Math Power 11 p.39 Technology Math Power 11 p.38, 39 Alg, Str & Method Book 2 p.792# 9-15 written ex p.804# 1,2,11-14 written ex 2. The local electric company s current rate schedule is $8.50 equipment charge and $.09 per Kwh of electricity used. Next year the company plans to charge a base fee of $6.00 and a $.10 per Kwh electricity charge. For what Kwh use-age will a home-owner s bill be the same for both years? Solution x = # Kwh s used y = amount of bill (Current Schedule) y = x (Next year s Schedule) y = x Journal Explain why the break-even point is important to business people. Extension Have advanced students design a situation such as the above two examples. 132

27 SCO: By the end of grade 11 students will be expected to: B18 solve systems of three or more equations using technology B9 use the calculator correctly and efficiently for various computations Elaboration- Instructional Strategies/Suggestions Three Dimensional Systems Invite students to experiment with 3-D systems in order to discover methods of solving this type of system. a) matrices Students may suggest using the TI-83 and matrices to solve. Use the same procedure as on p.25. An example is shown below: 4x + y + z = 5 2x!y +2z = 10 x!2y! z = 2 133

28 Three- Dimensional Systems Technology A farmer has 1200 hectares planted in potatoes, grain and corn. When the crops are planted, each hectare of potatoes requires 3 hours to plant, grain requires 1 hour and corn requires 2 hours for each hectare. Labour costs $12 per hour. Seed costs are $10 per hectare for potatoes, $15 per hectare for grain and $5 per hectare for corn. The farmer has planned to spend $24,000 on labour and $13,000 on seed. How many hectares of each crop can be planted? Solutions x = # hectares of potatoes planted y = # hectares of grain planted z = # hectares of corn planted # hectares planted x + y + z = 1200 seed costs 10x + 15y + 5z = labour costs 3(12)x + 1(12)y + 2(12)z = Three- Dimensional Systems Student Workbook # MathPower 11 p Alg, Str & Method Book 2 p.804 # 3-10 written ex Applications MathPower 11 p.45 #42-46,54 Research Talk to a number of farmers and find out the cost of seed for three different agricultural crops. Investigate the labour costs involved in these crops. Find out any other issues that are affecting farmers as to what crops they are planting. 134