The network diagram shows the bus routes that run between four cities, forming a network.
The arrows indicate the direction the buses travel. Arrows on both ends of an edge indicate that buses travel in both directions. Recall Matrix 𝐵 (shown) represents the bus routes.
1.
Label the rows and columns and Origination (From) and Destination (To) parts of the matrix.
2.
What is the value of 𝑟
2,3
? What does it represent in this situation?
3.
What is the value of 𝑟
2,3
∙𝑟
3,1
, and what does it represent in this situation?
4.
How many walks of length 2 are there from City 1 to City 1? From City 3 to City 2?
5.
Multiply B by itself; that is, square B. What do you notice about b
1,1
and b
3,2
? Explain, using what you know about matrix multiplication, why this occurred.

Name_________________________ Hour ______
The Purrrrrfect Cat Toy Company contacts your travel agency about available flights between their offices in Lansing, Akron, Orlando, Indianapolis, and Norfolk.
The graph below shows the flights going in and out of airports in Lansing, Akron,
Orlando, Indianapolis, and Norfolk.
1. Can you fly directly from Lansing to Akron? If not, what would be a possible route with layovers?
2. Can you fly directly from Orlando to Akron? If not, what would be a possible route with layovers?
3. What flight(s) depart from Norfolk?
4. What flight(s) arrive in Norfolk?
5. A matrix can be used to organize this graph. Create one that makes sense. Call it A .

6. What is
2
A ? What does it represent in the context of this problem?
7. What is
3
A ? What does it represent in the context of this problem?
8. An employee from the Purrrrrfect Cat Toy Company in Norfolk says that they want to go to Lansing and are willing to take up to 3 flights. a) How could you represent that using a matrix? b) What would the matrix be? c) How many routes with up to three flights are possible? List them. Do all of them make sense?
9. An employee in Orlando wants to go to the Akron office and will take up to three flights. How many routes are possible? Do all of them make sense?

Name________________________ Hour_____
Directed Graphs and Network Matrices Practice
1. Write the matrix for the directed graph given. Label the rows and columns.
2. Ship A sends messages to Ship B. Ship B sends messages and receives messages from Ships C and E. Ship D sends and receives messages from Ship C. Ship E receives messages from Ships A and C. a) Draw a network to represent this situation. b) Write the communication matrix M that represents this directed network. Label rows and columns alphabetically. c) Find M
2
. Interpret the meaning of the entry in the fifth row and second column.
3. There are n airports. Each airport has a direct flight to every other airport (besides itself). How many zeros are in the matrix? How many ones?

4. A dominance matrix is a special type of directed graph matrix that shows where dominance exists between two members. For example, if a certain chess player A defeats another player B
, they show “dominance” and there would be a “1” in the A row and B column. a) The table below shows the results of a chess club competition where each member played every other member in the club.
Player
Anne
Bill
Ciara
Dawn
Who they defeated
Dawn
Anne, Ciara, Dawn
Anne
Ciara, Evan
Evan Anne, Bill, Ciara
Create a dominance matrix to represent the tournament. Label the rows and columns in alphabetical order. b) What does the total of each row represent? What does the total of each column represent? c) One method to decide who comes in first, second, and third in a tournament is to look at the second-stage dominances. For example, Ciara has a second-stage dominance over Dawn since Ciara beat Anne and Anne beat Dawn. Find M
2
, which gives the second stage dominance. d) The person who has the greatest total of first- and second-stage dominances is the winner of the tournament.
Who finished first, second, and third in the tournament?