112 UNIT 2 • Matrix Methods Number of Toys Made Sept Oct Crabs 10 20 Ducks 25 30 Cows 10 30 Two of the contractors, Elise and Harvey, know from experience how many minutes it takes them to make each type of toy, as shown in this matrix: Time per Toy (in minutes) Crab Duck Cow Elise 55 60 90 Harvey 80 50 100 i. Use matrix multiplication to find a matrix that shows the total number of minutes each of the two contractors will need in order to fulfill their contracts for each of the two months. ii. Convert the minute totals to hours. What matrix operation could you use to do this conversion? Does your calculator or computer software have the capability to perform this type of matrix operation? b. Perform the following matrix multiplications without using a calculator or computer. Then check your answers using technology. i. 2 3 4 5 6 7 ii. 1 3 6 5 0 2 3 3 iii. 1 0 2 2 3 0 -4 1 1 2 3 4 0 1 2 iv. 2 5 1 3 x y v. 0 1 1 1 0 2 0 0 1 0 1 1 1 0 2 0 0 1 I I nvesti nvesti g g ation ation 3 3 The Power of a Matrix The Power of a Matrix In this investigation, you will learn a new way to represent and analyze information using matrices. As you explore problems about ecosystems and tennis tournaments, look for answers to these questions: How can you represent a vertex-edge graph with a matrix? If you multiply such a matrix by itself, what information do you get about the vertex-edge graph and the situation represented by the graph? CPMP-Tools Name: __________________________________ Unit 4 Lesson 2 Investigation 3
Transcript
112 UNIT 2 • Matrix Methods
Number of Toys Made
Sept Oct Crabs 10 20 Ducks 25 30 Cows 10 30
Two of the contractors, Elise and Harvey, know from experience how many minutes it takes them to make each type of toy, as shown in this matrix:
Time per Toy (in minutes)
Crab Duck Cow Elise 55 60 90 Harvey 80 50 100
i. Use matrix multiplication to find a matrix that shows the total number of minutes each of the two contractors will need in order to fulfill their contracts for each of the two months.
ii. Convert the minute totals to hours. What matrix operation could you use to do this conversion? Does your calculator or computer software have the capability to perform this type of matrix operation?
b. Perform the following matrix multiplications without using a calculator or computer. Then check your answers using technology.
i. 2 34 5
67
ii. 1 36 5
0 23 3
iii. 1 0 2 2 3 0 -4 1 1
2 34 01 2
iv. 2 51 3
xy
v.0 1 11 0 20 0 1
0 1 11 0 20 0 1
IInvest invest iggationation 33 The Power of a Matrix The Power of a MatrixIn this investigation, you will learn a new way to represent and analyze information using matrices. As you explore problems about ecosystems and tennis tournaments, look for answers to these questions:
How can you represent a vertex-edge graph with a matrix?
If you multiply such a matrix by itself, what information do you get about the vertex-edge graph and the situation represented by the graph?
CPMP-Tools
Name: __________________________________
Unit 4Lesson 2
Investigation 3
LESSON 2 • Multiplying Matrices 113
Pollution In an Ecosystem An ecosystem is the system formed by a community of organisms and their interaction with their environment. The diagram below shows the predator-prey relationships of some organisms in a willow forest ecosystem.
Willow Forest Ecosystem
spider
frog
snail gartersnake
bronzegrackle
yellowwarbler
meadowwillow
fleabeetlesawfly
Such a diagram is called a food web. An arrow goes from one organism to another if one is food for the other. So, for example, the arrow from spider to yellow warbler means that spiders are food for yellow warblers.
Pollution can cause all or part of the food web to become contaminated. In the following problems, you will explore how matrix multiplication can be used to analyze how contamination of some organisms spreads through the rest of the food web.
1 Using the willow forest ecosystem food web, discuss answers to the following questions.
a. How are predator-prey relationships represented in the food web diagram? What do the arrows mean?
b. Think about the effect on the ecosystem if a pollutant is introduced at some point in the forest. Assume the pollution does not kill any organisms, but the contamination is spread by eating.
i. What might happen when a toxic chemical washes into a stream in which the frogs live?
ii. What might happen if a pesticide contaminates the sawflies?
c. The food web can be viewed as a vertex-edge graph, where the vertices are the organisms and the edges are the arrows. Since the edges have a direction, this type of vertex-edge graph is sometimes called a directed graph, or digraph.
How can paths through the digraph help to analyze the spread of contamination? Illustrate your answer in the case where the contamination first effects the flea beetle.
114 UNIT 2 • Matrix Methods
2 Matrices can be used to help find paths through digraphs. The first step in finding paths is to construct an adjacency matrix for the food web digraph. You may recall from Core-Plus Mathematics Course 1 that an adjacency matrix is constructed by using the vertices of the digraph as labels for the rows and columns of a matrix. Each entry of the matrix is a “1” or a “0” depending on whether or not there is an arrow in the digraph (directed edge) from the row vertex to the column vertex.
a. Below is a partially completed adjacency matrix for the food web digraph. Complete the adjacency matrix by filling in all the blank entries. For consistency in this investigation, we will always list the organisms alphabetically across the columns and down the rows.
b. Compare your matrix with the matrices constructed by other students. Discuss and resolve any differences in your matrices so that everyone agrees upon the same matrix. Label this adjacency matrix A.
3 The adjacency matrix tells you if there is an edge from one vertex to another. An edge from one vertex to another is like a path of length one. Now think about paths of length two. A path of length two from one vertex to another means that you can get from one vertex to the other by moving along two consecutive directed edges.
a. The partially completed matrix below shows the number of paths of length two in the food web digraph. Complete the matrix.
b. Compare your matrix to those constructed by others. Discuss and resolve any differences so that everyone has the same matrix.
4 What matrix operation(s) could be used to get the paths-of-length-two matrix from the paths-of-length-one matrix A? Make and test some conjectures. (You may find it helpful to use vertex-edge graph software or other technology for this problem.)
5 Suppose that the meadow willows are contaminated by polluted ground water. In turn, they contaminate other organisms that feed directly or indirectly on them. However, at each step of the food chain, the concentration of contamination decreases.
a. Suppose that organisms more than two steps from the meadow willow in the food web are no longer endangered by the contamination. Using the digraph, find one organism that is safe.
b. How can the matrices be used to help find all the safe organisms? Explain your reasoning.
c. Compare your methods for finding all the safe organisms with others.
Tournament Rankings You have seen that powers of an adjacency matrix give you information about paths of certain lengths in the corresponding vertex-edge graph. This connection between graphs and matrices is useful for solving a variety of problems. For example, it is often very difficult to rank players or teams in a tournament accurately and systematically. A vertex-edge graph can give you a good picture of the status of the tournament. The corresponding adjacency matrix can help determine the ranking of the players or teams. Consider the following tournament situation.
The second round of a city tennis tournament involved six girls, each of whom was to play every other girl. However, the tournament was rained out after each girl had played only four matches. The results of play were the following:
• Erina beat Keadra.
• Akiko beat Julia.
• Keadra beat Akiko and Julia.
• Julia beat Erina and Maria.
• Maria beat Erina, Cora, and Akiko.
• Cora beat Erina, Keadra, and Akiko.
6 Using the information above, can you decide anything about how the girls should be ranked at this stage of the tournament? Explain your reasoning.
CPMP-Tools
116 UNIT 2 • Matrix Methods
Summarize the Mathematics
7 A digraph and an adjacency matrix can be used to help rank the girls at this stage of the tournament with no ties.
a. Represent the status of the tournament by completing a copy of the digraph and adjacency matrix below.
J
E
M
A
K
C
A C E J K M Akiko – – – – – – Cora – – – – – – Erina – – – – – – Julia – – – – – – Keadra – – – – – – Maria – – – – – –
b. Rank the girls as clearly as you can. Use the information shown in the digraph and adjacency matrix to explain your ranking.
c. If you did not use row sums of the adjacency matrix in Part b, what additional information do these sums provide?
d. Compute the square of the adjacency matrix and discuss what the entries tell you about the tournament. How could you use this information to help rank the girls?
e. Use further operations on the adjacency matrix to rank the players with no ties. Explain the ranking system you used. Compare your method and results with others.
In this investigation, you explored how powers of an adjacency matrix for a digraph and sums of the powers could be used to analyze the digraph and the situation it models.a Consider paths in a digraph.
i. How do paths in a food web help you track the spread of contamination through the ecosystem?
ii. What do paths in a tournament digraph tell you about the tournament?
b What do powers of the adjacency matrix tell you about paths in the digraph?
c Explain how you can use powers and sums of matrices to track pollution through an ecosystem and to rank the players in a tournament.
Be prepared to share your thinking and tournament-ranking plan with the class.
