<p> 1</p><p>Matrix Systems</p><p>[Assuming students know the basics of matrix multiplication]</p><p>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:</p><p>B</p><p>A D</p><p>C</p><p>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: ______</p><p>Project AMP A Quesada Director Project AMP 2</p><p>Do you have all the two-step paths? How do you know? ______</p><p>You should have thirteen two-step paths!</p><p>Using Matrices:</p><p>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.]</p><p>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.</p><p>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.</p><p>Let’s analyze the matrix M2:</p><p>Project AMP A Quesada Director Project AMP 3</p><p>What value do you have for A A? ______What do you think this number represents? ______</p><p>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? ______</p><p>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:</p><p>So we get:</p><p>To get the second row third column of the matrix M2 (the two-steps communication from B to C), we take:</p><p>Project AMP A Quesada Director Project AMP 4</p><p>So we get: </p><p>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.)</p><p>Project AMP A Quesada Director Project AMP 5</p><p>So the two-step path way we get is: team ______to team ______to team ______</p><p>Project AMP A Quesada Director Project AMP 6</p><p>Another two-step path we get for going from A and returning to A is:</p><p>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. </p><p>Now, add all the numbers in the M2 matrix. What did you get? ______What does this tell you about the M2 matrix? ______</p><p>Project AMP A Quesada Director Project AMP 7</p><p>______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.</p><p>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:</p><p>Team ______to team ______</p><p>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. </p><p>Project AMP A Quesada Director Project AMP 8</p><p>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? ______</p><p>Project AMP A Quesada Director Project AMP 9</p><p>From our stored matrices, we want to calculate M + M2. Use your calculator to find T, and enter your results in the figure below:</p><p>Look at the matrix T. What value do you have for A A ______</p><p>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? ______.</p><p>What is the relationship between matrices M, M2, and T? ______</p><p>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? ______</p><p>What do you think matrix T itself represents? Explain: ______</p><p>Project AMP A Quesada Director Project AMP 10</p><p>Project AMP A Quesada Director Project AMP 11</p><p>Now, use your calculator to find S, where S = M + M2 + M3. Write in the results below.</p><p>What does matrix S represent? Explain. ______Do you notice a big difference between matrices S and T? ______</p><p>Given a matrix such as M: The first matrix obtained as the sum of consecutive powers of M with no </p><p> zeros, is called the transitive closure of M, and denoted by M + .</p><p>List some practical reasons for the importance of knowing how many ways to connect a radio system: ______</p><p>Project AMP A Quesada Director Project AMP 12</p><p>Can you think of other uses for this type of matrix; especially with other types of connections? ______</p><p>Project AMP A Quesada Director Project AMP 13</p><p>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.</p><p>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? ______</p><p>Project AMP A Quesada Director Project AMP</p>