Assuming Students Know the Basics of Matrix Multiplication

Matrix Systems

[Assuming students know the basics of matrix multiplication]

Suppose you, A, and three of your friends, B, C, and D communicate with each other through the radio system. Your radio has 2 way communications with B and C [that is, you could call B and B could call you directly, same for C.]. C’s radio has a 2 way communication with D and one way communication with B. Look at the picture below:

Two Step Paths: Now, suppose we are interested in two-step paths [that is, we are interested in the number of calls in which someone acts as a “middle person” to relay communications from one person to another]. List all the two-step paths: ______

Project AMP A Quesada Director Project AMP 2

Do you have all the two-step paths? How do you know? ______

You should have thirteen two-step paths!

Using Matrices:

There is an easy mechanical way for answering this and similar questions, using the “incidence” matrix of a graph. Let’s learn how to create this matrix for the given graph. Look at the picture above to fill out the following matrix. Use 0 if the two people do not communicate directly. Also use 0 for a person communicating with himself/herself. Use 1 for direct communication from one person to another. [A A and A B have already been filled out for you, as examples.]

Analyze M 2 Now, using the TI83 calculator, enter in the matrix M. Use your calculator to find M2. Write in your results in the matrix below.

We want to store the matrix M2 for later usage, so push , 2nd, X-1, and choose a matrix (such as [B]), then press ENTER.

Let’s analyze the matrix M2:

What value do you have for A A? ______What do you think this number represents? ______

Let’s go back and look at your two-step paths. How many paths did you have going from A and returning to A? ______Does the number you have going from A and returning to A in the two-step paths match the number from A A on the M2 matrix? Why or why not? ______In conclusion, what does the number 2 in row one column one of the matrix M2 represent? ______

Quick Review: Now, remember your matrix multiplication: to get the first row and first column of the M2 matrix (two-steps communication from A to A), we take:

So we get:

To get the second row third column of the matrix M2 (the two-steps communication from B to C), we take:

So we get:

So when we multiply these matrices, we are really setting up a communications pathway. Each value within the matrix represents a certain number of communications between teams. Continue to Analyze the M2 Matrix. Look back at the diagram above for the communication going from A and returning to A again. The communication pathway we get is: (Since 0(0) is just 0, and 0(1) is just 0, we are not interested in them. We are interested in the one that gives us results.)

So the two-step path way we get is: team ______to team ______to team ______

Another two-step path we get for going from A and returning to A is:

The pathway we get is: team ______to team ______to team ______. Do these two paths match your answers in your two-step paths going from A to A? ______. Let’s look at the communications going from C to B. What number is in the M2 matrix for C B? ______So how many two-step path(s) communications should there be going from C to B? _____ Look at your two-step paths. Do you have the same number of possible communication line(s)? ______List the pathway in which the communication line travels from C to B using the M2 matrix. Show your work.

Now, add all the numbers in the M2 matrix. What did you get? ______What does this tell you about the M2 matrix? ______

______Analyze M 3 Now, let’s look at the M3 for this problem. Use your calculator to find M3, and then store it for later usage. Write your results in the matrix below.

Let’s analyze what information we can get from the matrix M3: What number do you have for A A in the matrix M3? ______List the pathway in which the communication line travels from A to A: Let’s start with the matrix M2 * M = M3. For three-step communications going from A to A, we get:

Team ______to team ______

Now, let’s analyze what we have. Which pair (numbers) above when multiplied, gives us results? ______Which number in the pair above belongs to the M2 matrix? ______. Now, we want to know how the number that belongs to the M2 matrix from above forms from M * M matrices.

Team ______to team ______Team ______to team ______Let’s put all the steps together: The pathway of the three-way communication going from A to A: team _____ to team ____ to team _____ to team ______Explain what the numbers in the M3 matrix represent. ______If you need to find the pathways represented by a number in matrix M3, how would you do it? Briefly describe your ideas. ______Based on what you have already discovered about matrices for this system, what kind of information do you think that a M n matrix would give you? ______

From our stored matrices, we want to calculate M + M2. Use your calculator to find T, and enter your results in the figure below:

Look at the matrix T. What value do you have for A A ______

Look at some of your earlier results: What was the value of A A for the matrix M? ______What was the value of A A for the matrix M2? ______.

What is the relationship between matrices M, M2, and T? ______

Look at the other numbers in matrix T and compare them with the corresponding values in matrices M and M2. In terms of what the matrices M and M2 represent, what do the numbers within in the matrix T represent? ______

What do you think matrix T itself represents? Explain: ______

Now, use your calculator to find S, where S = M + M2 + M3. Write in the results below.

What does matrix S represent? Explain. ______Do you notice a big difference between matrices S and T? ______

Given a matrix such as M: The first matrix obtained as the sum of consecutive powers of M with no

zeros, is called the transitive closure of M, and denoted by M + .

List some practical reasons for the importance of knowing how many ways to connect a radio system: ______

Can you think of other uses for this type of matrix; especially with other types of connections? ______

Problems: I. Suppose the water company wants to find out the water-line connections for 6 houses so they can figure out which water line to check first when the water is contaminated. They came up with a matrix that represents the direct water-line connection. 1 represents the direct connection, and 0 represents no direct connection. II.

1. Suppose the water in house B is contaminated. After checking the water-line of house B, the company wants to check the two-step water-line connections that start from house B to see if they are also contaminated. List all the two-step water-line connections that start from B: ______2. It takes ______-step water-line connections for all the water-lines of all 6 houses to all be connected. 3. How many one-step or two-step water-line connections are there? ______

Project AMP A Quesada Director Project AMP