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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:
B
A
D
C
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:
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Do you have all the two-step paths? How do you know?
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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.
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[A
A and A
B have already been filled out for you, as examples.]
Analyze M2
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
matrix (such as [B]), then press ENTER.
, 2nd , X-1 , and choose a
Let’s analyze the matrix M2:
What value do you have for A
A? _____________________
What do you think this number represents? _____________________________________
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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? ____________________
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In conclusion, what does the number 2 in row one column one of the matrix M2
represent?
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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:
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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 ________
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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? __________________________________
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Analyze M3
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.
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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.
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If you need to find the pathways represented by a number in matrix M3, how would you
do it? Briefly describe your ideas.
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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?
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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?
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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?
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What do you think matrix T itself represents? Explain:
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Now, use your calculator to find S, where S = M + M2 + M3.
Write in the results below.
What does matrix S represent? Explain. ____________________________________
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Do you notice a big difference between matrices S and T?
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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:
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Can you think of other uses for this type of matrix; especially with other types of
connections?
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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:
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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? ____________
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