Vocabulary. A network is a set of objects. that are connected together by a. The vertices of a network are the. The arcs or edges of a network show

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1 Unit 4+ Vocabulary Knowledge Rating* Lesson Vocabulary Terms Definition Picture/Example/Notation A network is a set of objects 1 Network that are connected together by a. 1 Vertices The vertices of a network are the. 1 Arcs or Edges The arcs or edges of a network show the between the objects. 1 Directed network graph If the arcs of the network have directed., then the network is If an arc of a network can go in 1 Bidirectional direction, then that arc is bidirectional. Knowledge Rating: N 5 I have no knowledge of the word. S 5 I ve seen the word, but I m not sure what it means. U 5 I understand this word and can use it correctly. 5

2 54 Module 2 Solving Equations and Systems of Equations Knowledge Rating* Lesson Vocabulary Terms Definition Picture/Example/Notation 2 Matrix A matrix is a numbers or expressions. array of The or expressions in 2 Elements of a matrix the matrix are called the elements. The subscript defines where in the matrix the element is located. For matrix A, a ij refers to the ith and the jth. 2 Dimension or Order of a matrix A matrix having m rows and n columns has dimension m n or order m n. 2 Square matrix A square matrix has the columns. number of rows and 2 Zero or Null matrix A zero or null matrix has element. for Knowledge Rating: N 5 I have no knowledge of the word. S 5 I ve seen the word, but I m not sure what it means. U 5 I understand this word and can use it correctly.

3 Unit 4+ Networks & Matrices Vocabulary 55 Knowledge Rating* Lesson Vocabulary Terms Definition Picture/Example/Notation Scalar multiplication Scalar multiplication is the result of multiplying every element by the same (scalar). If two matrices have the same Adding matrices dimensions, then you can add them by adding the elements in each row and column. If two matrices have the same Subtracting matrices dimensions, then you can subtract them by subtracting the elements in each row and column. When multiplying matrices, the number of of the first matrix 4 Multiplying matrices must be the same as the number of rows of the second matrix. To multiply matrices, the elements in each of the first matrix are multiplied by the elements in each column of the second matrix, then the products are. 5 Identity Matrix When the identity matrix, I, is with any square matrix the product is the original square matrix. The identity matrix has a main diagonal of s and all the other elements are 0 s. Knowledge Rating: N 5 I have no knowledge of the word. S 5 I ve seen the word, but I m not sure what it means. U 5 I understand this word and can use it correctly.

4 56 Module 2 Solving Equations and Systems of Equations Knowledge Rating* Lesson Vocabulary Terms Definition Picture/Example/Notation The determinant of matrix a b A = c d is found by the formula,. This is denoted as 6 Determinant det A or A. If the determinant is equal to 0, then inverse exists for that matrix. 6 Inverse matrix When you multiply a matrix by its inverse matrix, the product is the matrix. To find the inverse matrix, use the formula A 1 = 1 d deta c b a Knowledge Rating: N 5 I have no knowledge of the word. S 5 I ve seen the word, but I m not sure what it means. U 5 I understand this word and can use it correctly. Word Bank 0 1 ad 2 bc added arrows column columns connection corresponding corresponding either every identity multiplied no number numbers objects rectangular relationship row row same

5 Unit 4+ Networks & Matrices Vocabulary 57 Name: Period: Date: Vocabulary Activity Use the bubble map to answer the question, What are matrices? Write a vocabulary term in one of the bubbles to answer this question. For each connecting bubble write a single sentence to explain how that bubble is related to the essential question. Continue this process until you have all of the bubbles filled in. What are matrices?

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7 LESSON 1 Introduction to Networks Exploratory Challenge 1 one classic math puzzle is the Seven Bridges of Königsberg problem which laid the foundation for networks and graph theory. In the 18th century in the town of Königsberg, Germany, a favorite pastime was walking along the Pregel River and strolling over the town s seven bridges... A question arose: Is it possible to take a walk and cross each bridge only once? Source: Merian-Erben, 1652 You will need: highlighters 1. Work with your partner to devise a path that crosses each bridge only once. You may draw on the map above. 2. How many solutions did your class find? 59

8 540 Module 2 Solving Equations and Systems of Equations Leonhard Euler, a Swiss mathematician, proved in 176 that it was impossible to cross each bridge exactly once and go over every one of the seven bridges. He created a simplified version of the map so that only the bridges and land masses were visible. He then simplified this even more by using dots or vertices for the land masses and segments or arcs or edges for the bridges as shown below. This final map is called a network. rook76/shutterstock.com Sources: Merian-Erben, 1652 and and Use highlighters to show the connection between each model above. For example, you could highlight one bridge and its corresponding arcs in the same color. The ideas behind networks are found in many fields and occupations. Two different network examples are shown below. World Map with Global Technology Network Zsolt Biczo/Shutterstock.com

9 Unit 4+ Networks & Matrices Lesson 1 Introduction to Networks 541 Food Web Network Vecton/Shutterstock.com 4. Name one other network that you use or know of? Exploratory Challenge 2 In this exploration, your group will create your own network based on criteria about the bus routes to and from four cities. 5. Work with your group to draw a network for the bus routes in the space below. Be sure your finished network satisfies ALL the conditions for each city. Network Conditions The City 1 buses have routes to City 2 and City. The City 2 buses have one route to City 4. Network Conditions The City buses have routes to City 2 and City 4. There is no City 4 bus routes.

10 542 Module 2 Solving Equations and Systems of Equations The diagram you created in the Exercise 5 may have looked something like this: This is a vertex of the graph. This is an edge of the graph. 6. This is a called a directed graph. Why do you think it has this name? The routes from one city to another are edges on the graph and the cities are vertices. 7. How many ways can you travel from City 1 to City 4? Explain how you know. 8. What about these bus routes doesn t make sense?

11 Unit 4+ Networks & Matrices Lesson 1 Introduction to Networks 54 It turns out there was an error in printing the first route map. An updated network diagram showing the bus routes that connect the four cities is shown below. Arrows on both ends of an edge indicate that buses travel in both directions (bidirectional). 9. How many ways can you reasonably travel from City 4 to City 1 using the route map above? Explain how you know. A rival bus company offers more routes connecting these four cities as shown in the network diagram in at the right. Discussion 10. What might the loop at City 1 represent? 11. What might be difficult about describing the path from City 1 to City 4 with this diagram?

12 544 Module 2 Solving Equations and Systems of Equations To better analyze the different paths from one city to another, mathematicians often label each path. Because there are multiple routes to each city, we ll label the different routes with letters to distinguish one from another. In that way you can distinguish the path from City 1 to City 2 using Route A, Route B or Route C. 12. How many ways can you travel from City 1 to City 4 if you want to stop in City 2 and make no other stops? 1. How many possible ways are there to travel from City 1 to City 4 without repeating a city? Discussion As a transportation network grows, these diagrams become more complicated, and keeping track of all of the information can be challenging. People that work with complicated networks use computers to manage and manipulate this information. 14. What challenges did you encounter as you tried to answer Exercises 12 and 1? 15. How might we present the possible routes in a more organized manner?

13 Unit 4+ Networks & Matrices Lesson 1 Introduction to Networks 545 Lesson Summary A network is a collection of points, called vertices, and a collection of lines, called arcs or edges, connecting these points. A network is a graphical representation of a relationship between objects or ideas. This is a vertex of the graph. This is an edge of the graph. If the edges in a network are shown with arrows, then the network is called a directed network or directed graph. If no arrows appear in a network, then it is assumed that all edges are bidirectional. Directed Network Bidirectional Network

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15 Unit 4+ Networks & Matrices Lesson 1 Introduction to Networks 547 Name: period: Date: Homework Problem Set 1. Consider the railroad map between Cities 1, 2, and, as shown on the right. A. How many different ways can you travel from City 1 to City without passing through the same city twice? B. How many different ways can you travel from City 2 to City without passing through the same city twice? C. How many different ways can you travel from City 1 to City 2 with exactly one connecting stop? D. Why is this not a reasonable network diagram for a railroad? 2. Consider the subway map between stations 1, 2, and, as shown. A. How many different ways can you travel from station 1 to station without passing through the same station twice? B. How many different ways can you travel directly from station 1 to station with no stops? C. How many different ways can you travel from station 1 to station with exactly one stop? D. How many different ways can you travel from station 1 to station with exactly two stops? Allow for stops at repeated stations.

16 548 Module 2 Solving Equations and Systems of Equations. Consider the airline flight routes between Cities 1, 2,, and 4, as shown. A. How many different routes can you take from City 1 to City 4 with no stops? B. How many different routes can you take from City 1 to City 4 with exactly one stop? C. How many different routes can you take from City to City 4 with exactly one stop? D. How many different routes can you take from City 1 to City 4 with exactly two stops? Allow for routes that include repeated cities. E. How many different routes can you take from City 2 to City 4 with exactly two stops? Allow for routes that include repeated cities.

17 LESSON 2 Organizing Networks with Matrices Opening Exercise As we saw in Lesson 1, networks can become complicated and finding a way to organize that data is important. 1. We will consider a direct route to be a route from one city to another without going through any other city. organize the number of direct routes from each city into the table shown below. The first row showing the direct routes between City 1 and the other cities has been completed for you. DIRECT ROUTES Destination Cities Cities of Origin

18 550 Module 2 Solving Equations and Systems of Equations A matrix is defined as a rectangular array of numbers arranged in the form shown. a a a a a a a a a n n m1 m2 mn 2. Use the table on the previous page to represent the number of direct routes between the four cities in matrix R. R = For our table of Direct Routes, a matrix would only include the inner cells and not the City labels (1, 2, and 4). Elements: The numbers in the matrix (e.g. a 12, a 1n and a mn ) are called the elements. The subscript defines where in the matrix the element is located. The first number determines the row and the second determines the column.. Circle r 2,. What is the value of r 2,? What does it represent in this situation? Order or Dimension of the Matrix: A matrix having m rows and n columns is called a m n (m by n) matrix. A m n matrix has order m n. 4. What is the order of matrix R? Square Matrix: An n n matrix is a square matrix since the number of rows is equal to the number of columns. 5. Is matrix R a square matrix? How can you tell?

19 Unit 4+ Networks & Matrices Lesson 2 Organizing Networks with Matrices 551 Null or Zero Matrix: A null or zero matrix is a matrix with all elements zero. 6. What would it mean if matrix R was a zero matrix? What would that represent in real life? Exploratory Challenge 7. What is the value of r 2, r,1, and what does it represent in this situation? 8. A. Write an expression for the total number of one-stop routes from City 4 to City 1. B. Determine the total number of one-stop routes from City 4 to City Do you notice any patterns in the expression for the total number of one-stop routes from City 4 to City 1? 10. How can you find the total number of possible routes between two locations in a network?

20 552 Module 2 Solving Equations and Systems of Equations Working Backward Going from a Matrix to a Network 11. Create a network diagram for the matrices shown below. Each matrix represents the number of transportation routes that connect four cities. The rows are the cities you travel from, and the columns are the cities you travel to. T = U =

21 Unit 4+ Networks & Matrices Lesson 2 Organizing Networks with Matrices 55 Arc Diagrams Here is a type of network diagram called an arc diagram. Notice that there are no arrows on this diagram. When there are no arrows, the arcs are bidirectional. Suppose the points represent eleven students in your mathematics class, numbered 1 through 11. The arcs above and below the line of vertices 1 11 are the people who are friends on a social network. 12. Complete the matrix that shows which students are friends with each other on this social network. The first row has been completed for you. Student 1 is friends with Student 1 is not friends with Student 10. How many ways could Student 1 get a message to 10 by only going through one other friend? 14. Who has the most friends in this network? Explain how you know. 15. Is everyone in this network connected at least as a friend of a friend? Explain how you know.

22 554 Module 2 Solving Equations and Systems of Equations Lesson Summary A matrix is a rectangular array of numbers, symbols or expressions, arranged in rows and columns. The individual items in a matrix are called its elements. Complete the matrix for this network

23 Unit 4+ Networks & Matrices Lesson 2 Organizing Networks with Matrices 555 Name: period: Date: Homework Problem Set 1. Consider the railroad map between Cities 1, 2, and, as shown. Create a matrix R to represent the railroad map between Cities 1, 2, and. 2. Consider the subway map between stations 1, 2, and, as shown. Create a matrix S to represent the subway map between stations 1, 2, and.

24 556 Module 2 Solving Equations and Systems of Equations. Suppose the matrix R represents a railroad map between cities 1, 2,, 4, and 5. R = ben Bryant/Shutterstock.com A. How many different ways can you travel from City 1 to City with exactly one connection? B. How many different ways can you travel from City 1 to City 5 with exactly one connection? C. How many different ways can you travel from City 2 to City 5 with exactly one connection?

25 Unit 4+ Networks & Matrices Lesson 2 Organizing Networks with Matrices Let B = represent the bus routes between cities. A. Draw an example of a network diagram represented by this matrix. B. How many routes are there between City 1 and City 2 with one stop in between? C. How many routes are there between City 2 and City 2 with one stop in between? D. How many routes are there between City and City 2 with one stop in between?

26 558 Module 2 Solving Equations and Systems of Equations 5. Consider the following directed graph representing the number of ways Trenton can get dressed in the morning (only visible options are shown): A. What reasons could there be for there to be three choices for shirts after traveling to shorts but only two after traveling to pants? B. What could the order of the vertices mean in this situation? C. Write a matrix A representing this directed graph. D. Delete any rows of zeros in matrix A, and write the new matrix as matrix B. Does deleting this row change the meaning of any of the entries of B? If you had deleted the first column, would the meaning of the entries change? Explain. E. Calculate b 1,2 b 2,4 b 4,5. What does this product represent? F. How many different outfits can Trenton wear assuming he always wears a watch?

27 LESSON Networks and Matrix Arithmetic Opening Exercise Suppose a subway line also connects the four cities. Here is the subway and bus line network. the bus routes connecting the cities are represented by solid lines, and the subway routes are represented by dashed arcs. 1. Write a matrix B to represent the bus routes and a matrix S to represent the subway lines connecting the four cities. B = S = Exploratory Challenge Use the network diagram from the opening Exercise and your two matrices to help you complete this challenge with your group. 2. Suppose the number of bus routes between each city were doubled. A. What would the new bus route matrix be? B. Mathematicians call this matrix 2B. Why do you think they call it that? 559

28 560 Module 2 Solving Equations and Systems of Equations. A. What would be the meaning of 10B in this situation? B. Write the matrix 10B. C. How would you describe the process you used to create the matrix 10B? 4. Suppose we ignore whether or not a line connecting cities represents a bus or subway route. A. Create one matrix that represents all the routes between the cities in this transportation network. B. Why would it be appropriate to call this matrix B 1 S? Explain your reasoning. C. Why would B and S have to be the same dimensions in order to find their sum?

29 Unit 4+ Networks & Matrices Lesson Networks and Matrix Arithmetic Suppose that April s Pet Supply has three stores in Cities 1, 2, and. Ben s Pet Mart has two stores in Cities 1 and 2. Each shop sells the same type of dog crates in size 1 (small), 2 (medium), (large), and 4 (extra large). April s and Ben s inventory in each city are shown in the tables below. Bussakorn Ewesakul/Shutterstock.com April s Pet Supply Ben s Pet Mart City 1 City 2 City City 1 City 2 City Size Size Size Size Size Size Size Size A. Create a matrix A so that a i, j represents the number of crates of size i available in April s store j. B. Explain how the matrix B = Mart can represent the dog crate inventory at Ben s Pet C. Suppose that April and Ben merge their inventories. Write a matrix that represents their combined inventory in each of the three cities.

30 562 Module 2 Solving Equations and Systems of Equations Lesson Summary Complete the example for each operation. Matrix Operations Operation Symbols How to Calculate Examples Scalar Multiplication ka Multiply each element of matrix A by the real number k = The Sum of Two Matrices A 1 B Add corresponding elements in each row and column of A and B = Matrices A and B MUST have the same dimensions. The Difference of Two Matrices A 2 B 5 A 1 (21)B Subtract corresponding elements in each row and column of A and B = The matrices MUST have the same dimensions.

31 Unit 4+ Networks & Matrices Lesson Networks and Matrix Arithmetic 56 Name: period: Date: Homework Problem Set For the matrices given below, perform each of the following calculations or explain why the calculation is not possible. A = B = C = d = A 1 B 2. 2A 2 B. A 1 C 4. 22C 5. 4D 2 2C 6. B 2 B 7. 5B 2 C 8. B 2 A

32 564 Module 2 Solving Equations and Systems of Equations 2 9. Let A = 1 5 and B = A. If C 5 6A 1 6B, determine matrix C. B. If D 5 6(A 1 B), determine matrix D. C. What is the relationship between matrices C and D? Why do you think that is? Let A = 1 5 and X be a 2 matrix. If A+ X = , then determine X. 11. Let A = three cities and B = represent the bus routes of two companies between A. Let C 5 A 1 B. Find matrix C. Explain what the resulting matrix and entry c 1, mean in this context. B. Let D 5 B 1 A. Find matrix D. Explain what the resulting matrix and entry d 1, mean in this context. C. What is the relationship between matrices C and D? Why do you think that is?

33 LESSON 4 Understanding Matrix Multiplication Opening Exercise The subway and bus line network connecting four cities that we used in Lesson 2 is shown at the right. The bus routes connecting the cities are represented by solid lines, and the subway routes are represented by dashed lines. 1. Suppose we want to travel from City 2 to City 1, first by bus and then by subway, with no more than one connecting stop. A. Complete the chart below showing the number of ways to travel from City 2 to City 1 using first a bus and then the subway. The first row has been completed for you. First Leg (BUS) Second Leg (SUBWAY) Total Ways to Travel City 2 to City 1: 2 City 1 to City 1: City 2 to City 2: City 2 to City 1: City 2 to City : City to City 1: City 2 to City 4: City 4 to City 1: 565

34 566 Module 2 Solving Equations and Systems of Equations B. How many ways are there to travel from City 2 to City 1, first on a bus and then on a subway? How do you know? C. Why are the total ways to travel between City 2 to City 1 through City 1 equal to 0? Exploratory Challenge: The Meaning of Matrix Multiplication Suppose we want to travel between all cities, traveling first by bus and then by subway, with no more than one connecting stop. 2. Use a chart like the one in the Opening Exercise to help you determine the total number of ways to travel from City 1 to City 4 using first a bus and then the subway. First Leg (BUS) Second Leg (SUBWAY) Total Ways to Travel The total number of ways to travel from City 1 to City 4 by bus and then by subway:

35 Unit 4+ Networks & Matrices Lesson 4 Understanding Matrix Multiplication 567. Suppose we create a new matrix P to show the number of ways to travel between the cities, first by bus and then by subway, with no more than one connecting stop. A. Record your answers to Opening Exercises, Part B and Exercise 2 in the matrix below in the appropriate row and column location. We do not yet have enough information to complete the entire matrix. P = B. Explain how you decided where to record these numbers in the matrix. 4. What is the total number of ways to travel from City to City 2 first by bus and then by subway with no more than one connecting stop? Explain how you got your answer and where you would record it in matrix P.

36 568 Module 2 Solving Equations and Systems of Equations Matrix B, below, shows the number of bus lines connecting the cities in this transportation network, and matrix S, represents the number of subway lines connecting the cities in this transportation network. B = and S = What does the product b 1,2 s 2,4 represent in this situation? What is the value of this product? 6. What does b 1,4 s 4,4 represent in this situation? What is the value of this product? Does this make sense? 7. Calculate the value of the expression b 1,1 s 1,4 1 b 1,2 s 2,4 1 b 1, s,4 1 b 1,4 s 4,4. What is the meaning of this expression in this situation? 8. Circle the first row of B and the fourth column of S. How are these entries related to the expression above and your work in Exercise 2?

37 Unit 4+ Networks & Matrices Lesson 4 Understanding Matrix Multiplication A. Write an expression that represents the total number of ways you can travel between City 2 and City 1, first by bus and then by subway, with no more than one connecting stop. B. What is the value of this expression? C. What is the meaning of the result? D. Where in matrix P would you put this value?

38 570 Module 2 Solving Equations and Systems of Equations 10. A. Write an expression that represents the total number of ways you can travel between City 4 and City 1, first by bus and then by subway, with no more than one connecting stop. B. What is the value of this expression? C. What is the meaning of the result? D. Where in matrix P would you put this value?

39 Unit 4+ Networks & Matrices Lesson 4 Understanding Matrix Multiplication 571 Discussion 11. A. What does each element of matrix P represent? B. What patterns do you notice in the expressions in Exercises 7, 9 and 10? C. Complete the sentence: To calculate the element of matrix P in the 2nd row and 4th column, you would... D. Complete the sentence: The element of matrix P in the 4th row and 2nd column represents the number of way to travel... E. Describe how to calculate any element in matrix P.

40 572 Module 2 Solving Equations and Systems of Equations 12. Complete matrix P that represents the routes connecting the four cities if you travel first by bus and then by subway. P = Lesson Summary Operation Symbols How to Calculate Example To find the element in the ith row and jth column of the product matrix, 1 0 multiply corresponding elements from the ith row Matrix A B of the first matrix by the Multiplication jth column of the second matrix, and then add the results. Repeat this process for each element in the product matrix When we multiply two matrices together, such as an m n matrix by an n p matrix, what is the size of the resulting matrix? = The resulting matrix has size.

41 Unit 4+ Networks & Matrices Lesson 4 Understanding Matrix Multiplication 57 Name: period: Date: Homework Problem Set 1 1. Let A = 2 0 and B 1 2 = represent the bus 4 routes of two companies between two cities. Find the product A B, and explain the meaning of the entry in row 1, column 2 of A B in the context of this scenario. parose/shutterstock.com 2. Let A = three cities and B = represent the bus routes of two companies between A. Let C 5 A B. Find matrix C, and explain the meaning of entry c 1,. B. Nina wants to travel from City to City 1 and back home to City by taking a direct bus from Company A on the way to City 1 and a bus from Company B on the way back home to City. How many different ways are there for her to make this trip? C. Oliver wants to travel from City 2 to City by taking first a bus from Company A and then taking a bus from Company B. How many different ways can he do this? D. How many routes can Oliver choose from if travels from City 2 to City by first taking a bus from Company B and then taking a bus from Company A?

42 574 Module 2 Solving Equations and Systems of Equations. Consider the matrices A = and B = Multiply AB and BA or explain why you cannot. For the matrices given below, perform each of the following calculations or explain why the calculation is not possible. A = B = C = D =

43 Unit 4+ Networks & Matrices Lesson 4 Understanding Matrix Multiplication AB 5. BC 6. AC 7. AD 8. A 2 9. C A 1 B 11. B 1 BC 12. Let F be an m n matrix. Then what do you know about the dimensions of matrix G in the problems below if each expression has a value? A. F 1 G B. FG C. GF

44 576 Module 2 Solving Equations and Systems of Equations 1. Consider an m n matrix A such that m n. Explain why you cannot evaluate A Let A = 2 0 1, B = 1 0 1, C = A, B, and C between three cities A. Zane wants to fly from City 1 to City by taking Airline A first and then Airline B second. How many different ways are there for him to travel? represent the routes of three airlines Olena Yakobchuk/Shutterstock.com B. Zane did not like Airline A after the trip to City, so on the way home, Zane decides to fly Airline C first and then Airline B second. How many different ways are there for him to travel?

45 LESSON 5 Properties of Matrices Opening Exercise In Lesson 4, we used the subway and bus line network connecting four cities. The bus routes connecting the cities are represented by solid lines, and the subway routes are represented by dashed lines. matrix B, below, shows the number of bus lines connecting the cities in this transportation network, and matrix S, represents the number of subway lines connecting the cities in this transportation network. B = We found that B S 5 P as shown below. and S = B i S = i = = p Calculate matrix M that represents the routes connecting the four cities if you travel first by subway and then by bus. 577

46 578 Module 2 Solving Equations and Systems of Equations 2. Should these two matrices (P and M) be the same? Explain your reasoning. We ve shown that matrix multiplication is generally not commutative, meaning that as a general rule for two matrices A and B, A B B A.. Explain why F G 5 G F in each of the following examples. A. F = 1 G 2 0, 2 6 = B. F = 1 2, G = C. F = 1 2, G = D. F = 1 2, G =

47 Unit 4+ Networks & Matrices Lesson 5 Properties of Matrices 579 Suppose each city had a trolley car that ran a route between tourist destinations. The double dotted loops represent the trolley car routes. Remember that straight lines indicate bus routes, and dotted lines indicate subway routes. 4. Explain why the matrix I shown below would represent the number of routes connecting cities by trolley car in this transportation network. I = Recall that B is the bus route matrix. Show that I B 5 B. Explain why this makes sense in terms of the transportation network. B = The real number 1 has the property that 1 a 5 a for all real numbers a, and we call 1 the multiplicative identity. Why would mathematicians call I an identity matrix? 7. What would be the form of a 2 2 identity matrix? What about a identity matrix?

48 580 Module 2 Solving Equations and Systems of Equations Let A = A. Construct a matrix Z such that A 1 Z 5 A. Explain how you got your answer. B. Explain why k Z 5 Z for any real number k. C. The real number 0 has the properties that a a and a for all real numbers a. Why would mathematicians call Z a zero matrix? 9. In this lesson you learned that the commutative property does not hold for matrix multiplication. This exercise asks you to consider other properties of real numbers applied to matrix arithmetic. A. Is matrix addition associative? That is, does (A 1 B) 1 C 5 A 1 (B 1 C) for matrices A, B, and C that have the same dimensions? Explain your reasoning.

49 Unit 4+ Networks & Matrices Lesson 5 Properties of Matrices 581 B. Is matrix multiplication associative? That is, does (A B) C 5 A (B C) for matrices A, B, and C for which the multiplication is defined? Explain your reasoning. C. Is matrix addition commutative? That is, does A 1 B 5 B 1 A for matrices A and B with the same dimensions?

50 582 Module 2 Solving Equations and Systems of Equations Lesson Summary Identity Matrix: The n n identity matrix is the matrix whose entry in row i and column i is 1, and all other entries are all zero. The identity matrix is denoted by I. The 2 2 identity matrix is 0 1 0, and the identity matrix is If the size of the identity matrix is not explicitly stated, then the size is implied by context.. Zero Matrix: The m n zero matrix is the m n matrix in which all entries are equal to zero. 0 For example, the 2 2 zero matrix is, and 0 the zero matrix is If the size of the zero matrix is not specified explicitly, then the size is implied by context.

51 Unit 4+ Networks & Matrices Lesson 5 Properties of Matrices 58 Name: period: Date: Homework Problem Set 1. Consider the matrices. A = and B = A. Would you consider B to be an identity matrix for A? Why or why not? B. Would you consider I = or I = an identity matrix for A? Why or why not? Recall the bus and trolley matrices from the lesson: B = and I = A. Explain why it makes sense that BI 5 IB in the context of the problem. B. Multiply out BI and show BI 5 IB. C. Consider the multiplication that you did in Part B. What about the arrangement of the entries in the identity matrix causes BI 5 B?

52 584 Module 2 Solving Equations and Systems of Equations. Let A = one airline between 4 cities. represent airline flights of A. We use the notation A 2 to represent the product A A. Calculate A 2. What do the entries in matrix A 2 represent? Ekaphon maneechot/shutterstock.com B. Jade wants to fly from City 1 to City 4 with exactly one stop. How many different ways are there for her to travel? C. Now Jade wants to fly from City 1 to City 4 with exactly two stops. How many different ways are there for her to choose?

53 LESSON 6 Using Matrices to Solve Systems of Equations Opening Exercise In the last unit, you learned to solve systems of equations by graphing, and by the substitution and elimination methods. Systems of equations can also be solved using matrices. Before we use matrices, let s use a simple example to review solving systems of equations. 1. Solve the system x 2 y = 4 by graphing and then use either the substitution method or x + y = 5 the elimination method to verify your results. Graphing Method x 2y = 4 x + y = 5 Substitution or Elimination Method x 2y = 4 x + y = 5 The solution to x 2 y = 4 is x and y. x + y = 5 585

54 586 Module 2 Solving Equations and Systems of Equations To solve the system, x 2 y = 4, with matrices, we have to see this system as three matrices as x + y = 5 shown below. 1 2 x 4 i 1 y = 5 2. What does each matrix represent? 1 2 x. Determine 1 i. How does it relate to the original equation? y To solve for the matrix x 1 2, we need to use some type of operation to move the matrix y 1 to the other side of the equation. In algebra we use inverse operations to solve equations and we ll do the same here. We need the inverse of the matrix When you multiply a matrix, A, by its inverse, A 21, you get the identify matrix, I. AA 21 5 A 21 A 5 I Suppose A, X and B are matrices and AX 5 B, then AA 21 X 5 A 21 B. This gives I X 5 A 21 B or X 5 A 21 B

55 Unit 4+ Networks & Matrices Lesson 6 Using Matrices to Solve Systems of Equations 587 To determine the inverse, A 21, you ll need a special number called the determinant, and it is written as A or det A. If A = a b c d, then the determinant of A or det A is given by the formula A 5 ad 2 bc. 4. Find the determinant of the matrix To find the inverse you ll use the reciprocal of the inverse and then manipulate the matrix. The steps are given below. If A = a b c d and A 5 ad 2 bc, then A 1 = 1 A d c b a. 5. If the determinant of the matrix is 0, then no inverse exists. Why is that so? 6. What is the inverse of matrix 1 2 1? (We ll call this Matrix A from here on.)

56 588 Module 2 Solving Equations and Systems of Equations 7. Solve the system using A x 4 i 1 y = 5 Write A 21 here. And here. 1 2 x 4 i i i 1 y = 5 8. Does your answer agree with your work in Exercise 1? 9. Which method, graphing, substitution, elimination or matrices do you like best? Why?

57 Unit 4+ Networks & Matrices Lesson 6 Using Matrices to Solve Systems of Equations 589 We ve been looking at transportation problems throughout this unit, but most transportation problems are so complex that you need computers or graphing calculators to solve them. Most require at least 4 4 matrices. We ll look at some typical 2-variable system problems which will give us 2 2 matrices. 10. Admission to the county fair is $2 for children and $5 for adults. Last Sunday, 1900 people came to the fair and they collected $5900 at the entrance booth. Kobby Dagan/Shutterstock.com A. If C represents the number of children who attended and A represents the number of adults, what are the two equations to represent this situation? B. Write the matrix equation that represents this system. C. Find the determinant and the inverse, and then solve the system with matrices. D. Number of children: Number of adults:

58 590 Module 2 Solving Equations and Systems of Equations 11. The sum of two numbers is 1 and their difference is 15. A. Write a system of equations for the two numbers. Be sure to define the variables you are using. B. Solve the system using matrices. Allies Interactive/Shutterstock.com

59 Unit 4+ Networks & Matrices Lesson 6 Using Matrices to Solve Systems of Equations 591 Lesson Summary You can use matrices to solve systems of equations. Complete the example below by following the steps given. Steps Explained Example: Solve 2x y = 8 x + 2y = 7 using matrices 1. Write the matrix equation. 2. Find the determinant.. Determine the inverse matrix. 4. Use the inverse to isolate the variable matrix. 5. Multiply to solve for the variables. 6. Check your answer.

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61 Unit 4+ Networks & Matrices Lesson 6 Using Matrices to Solve Systems of Equations 59 Name: period: Date: Homework Problem Set 1. Solve the system, x 2 y = 1 using matrices. x + 4y = 8 In Exercise 5, you explained why matrices with a determinant of 0 has no inverse. Find the determinant of each matrix and then decide which have no inverse

62 594 Module 2 Solving Equations and Systems of Equations 6. Julie went to the Taco Truck and bought 5 tacos and 2 burritos for $ Kent bought tacos and 4 burritos for $ Use matrices to determine how much each taco costs. Artisticco/Shutterstock.com

63 UNIT 4+ Reference Lesson 1 Introduction to Networks Lesson Summary A network is a collection of points, called vertices, and a collection of lines, called arcs or edges, connecting these points. A network is a graphical representation of a relationship between objects or ideas. This is a vertex of the graph. This is an edge of the graph. If the edges in a network are shown with arrows, then the network is called a directed network or directed graph. If no arrows appear in a network, then it is assumed that all edges are bidirectional. Directed Network Bidirectional Network 595

64 596 Module 2 Solving Equations and Systems of Equations Lesson 2 Organizing Networks with Matrices Elements: The numbers in the matrix (e.g. a 12, a 1n and a mn ) are called the elements. The subscript defines where in the matrix the element is located. The first number determines the row and the second determines the column. Order or Dimension of the Matrix: A matrix having m rows and n columns is called a m n (m by n) matrix. A m n matrix has order m n. Square Matrix: An n n matrix is a square matrix since the number of rows is equal to the number of columns. Null or Zero Matrix: A null or zero matrix is a matrix with all elements zero. Lesson Summary A matrix is a rectangular array of numbers, symbols or expressions, arranged in rows and columns. The individual items in a matrix are called its elements. Complete the matrix for this network

65 Unit 4+ Networks & Matrices Reference 597 Lesson Networks and Matrix Arithmetic Lesson Summary Matrix Operations Operation Symbols How to Calculate Examples Scalar Multiplication ka Multiply each element of matrix A by the real number k = Add corresponding elements in The Sum of Two Matrices A 1 B each row and column of A and B. Matrices A and B MUST have the same dimensions = Subtract corresponding elements in each The Difference of Two Matrices A 2 B 5 A 1 (21)B row and column of A and B. The matrices MUST have the same dimensions =

66 598 Module 2 Solving Equations and Systems of Equations Lesson 4 Understanding Matrix Multiplication Lesson Summary Operation Symbols How to Calculate Example To find the element in the jth row and ith column of the product matrix, multiply corresponding elements from the ith row Matrix A B of the first matrix by the Multiplication 6 12 jth column of the second matrix, and then add the results. Repeat this process for each element in the product matrix. When we multiply two matrices together, such as an m n matrix by an n p matrix, what is the size of the resulting matrix? = The resulting matrix has size m p.

67 Unit 4+ Networks & Matrices Reference 599 Lesson 5 Properties of Matrices Lesson Summary Identity Matrix: The n n identity matrix is the matrix whose entry in row i and column i is 1, and all other entries are all zero. The identity matrix is denoted by I. The 2 2 identity matrix is , and the identity matrix is If the size of the identity matrix is not explicitly stated, then the size is implied by context. Zero Matrix: The m n zero matrix is the m n matrix in which all entries are equal to zero. For example, the 2 2 zero matrix is , and the zero matrix is If the size of the zero matrix is not specified explicitly, then the size is implied by context.

68 600 Module 2 Solving Equations and Systems of Equations Lesson 6 Using Matrices to Solve Systems of Equations When you multiply a matrix, A, by its inverse, A 21, you get the identify matrix, I. AA 21 5 A 21 A 5 I Suppose A, X and B are matrices and AX 5 B, then AA 21 X 5 A 21 B. This gives I X 5 A 21 B or X 5 A 21 B To determine the inverse, A 21, you ll need a special number called the determinant, and written as A or det A. If A = a b c d, then the determinant of A or det A is given by the formula A 5 ad 2 bc. To find the inverse you ll use the reciprocal of the inverse and then manipulate the matrix. The steps are given below. If A = a b c d and A 5 ad 2 bc, then A 1 = 1 A d c b a.

69 Unit 4+ Networks & Matrices Reference 601 Lesson Summary You can use matrices to solve systems of equations. Complete the example below by following the steps given. Steps Explained Example: Solve = + = x y x y using matrices 1. Write the matrix equation. x y i = 2. Find the determinant. 2 i 2 2 (21) i (21) Determine the inverse matrix = 4. Use the inverse to isolate the variable matrix. x y x y i i i i = = 5. Multiply to solve for the variables. x y x y ( 8) ( 8) i i i i i = = + + = + + = = 6. Check your answer. 2( ) 2 8 ( ) 2(2) 7 = + =

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71 UNIT 4+ Matrices Appendix Using the Graphing Calculator for Matrix Operations Basic Matrix Manipulation with a Graphing Calculator Department of Mathematics, Sinclair Community College, Dayton, OH Often, a matrix may be too large or too complex to manipulate by hand. For these types of matrices, we can employ the help of graphing calculators to solve them. We will cover a few of the most common graphing calculators used in education today. Throughout the directions, calculator buttons with arrows indicate the operation order. To learn more, or if your calculator is not demonstrated, consult the manufacturer s product manual. I will be using the TI 8 Plus graphing calculator for these directions. The TI 84 Plus family of graphing calculators is the upgraded version of the TI 8 Plus, with possible extra features in the menus demonstrated below. However, the older TI 8 and TI 82 graphing calculators have slightly different keyboard layouts than the TI 8 Plus, so these directions will not be entirely accurate for those calculators. For example, instead of pressing 2nd MATRIX4 to access the matrix menu, just press the dedicated MATRIX button. Inputting/Editing Matrices: 60

72 604 Module 2 Solving Equations and Systems of Equations Adding and Subtracting Matrices:

73 Unit 4+ Networks & Matrices Matrices Appendix 605 Multiplying Matrices: Calculating the Inverse: Determinants:

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