Operations with Matrices Dan Teague NC School of Science and Mathematics [email protected].
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Transcript of Operations with Matrices Dan Teague NC School of Science and Mathematics [email protected].
Operations with Matrices
Dan TeagueNC School of Science and Mathematics
Reasoning and Sense-Making
One key is to find a way of introducing new material that fits into the jigsaw puzzle created by what is already understood.
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Working with Units
If you are walking 2 miles per hour, how fast are you traveling in feet per hour?
We know there are 5,280 feet in a mile, so
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feet miles feet5,280 2 10,560
mile hour hour
If you are walking 2 miles per hour, how fast are you traveling in miles per minute?
We know there are 60 minutes in an hour, so
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miles minutes2 60 ?hour hour
miles 1 hours 1 miles2hour 60 minute 30 minute
Think about the Mathematical Practices
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Now, on to Matrices
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What is a Matrix?
A matrix is a rectangular array of numbers.
A matrix as a model is a rectangular array of numbers with meaning.
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The Summer Woodworking
Project
You and a Partner make 3 kinds of cutting boards:
Type I: alternating Oak and Walnut Strips
Type II: alternating Oak, Walnut, Cherry Strips
Type III: checkerboard pattern of Walnut and Cherry
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Creating a Matrix
During weekends in the month of April, you constructed 5 Type I, 8 Type II, and 3 Type III cutting boards. Your partner constructed 6 Type I, 4 Type II, and 10 Type III cutting boards.
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You
Part
5 8 3
6 4 10
I II III
A
1,2A
April and May Results
Dimensions of the MatricesInterpretation of the Elements
You
Part
5 8 3
6 4 10
I II III
A
You
Part
4 12 5
6 10 8
I II III
M
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How many type II’s did your partner make in the 2 months?
How many type I’s did you make in the 2 months?
Can you create a matrix representing the work done by each person during the 2 months?
You
Part
5 8 3
6 4 10
I II III
A
You
Part
4 12 5
6 10 8
I II III
M
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You
Part
5 8 3
6 4 10
I II III
A
You
Part
4 12 5
6 10 8
I II III
M
You
Part
I II III
S
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Define this process as Addition
Notice the order. We didn’t define addition then practice. We asked what made sense, and used that for our definition.
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How many of each type did the partners make during June?
You
Part
20 22 15
16 21 16
I II III
S J
You
Part
9 20 8
12 14 18
I II III
S
You
Part
I II III
J
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Now We Can Define Subtraction
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S J S J
A M M A
Next Spring you want to Doubled your April Output
You
Part
5 8 3
6 4 10
I II III
A
2A A A http://courses.ncssm.edu/math/talks/conferences/
Types and Wood
Interpret the elements in the Wood matrix.
8 9 0
6 6 8
4 10 10
o w cI
II
III
W
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In April, how much Oak did You use?
In April, how much Cherry did the Partner use?
You
Part
5 8 3
6 4 10
I II III
A
8 9 0
6 6 8
4 10 10
o w cI
II
III
W
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8 9 0
6 6 8
4 10 10
o w cI
II
III
W
You
Part
5 8 3
6 4 10
I II III
A
You
Part
100
132
o w c
T
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8 9 0
6 6 8
4 10 10
o w cI
II
III
W
You
Part
5 8 3
6 4 10
I II III
A
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Define Matrix Multiplication
Just like (ft/mile)∙(miles/hr) = ft/hr
n m m k n kA B C
Person Type Type Wood Person WoodA B C
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Cost Matrix
Each strip of Oak costs 2 cents.Each strip of Walnut costs 4 cents.Each strip of Cherry costs 5 cents.
What is the cost of the wood used by each person?
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Cost Matrix
Oak costs 2 cents/strip, Walnut costs 4 cents/strip, Cherry costs 5 cents/strip.
$
You You
Part Part
100 123 94
112 178 132
o w c
T Z
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You
Part
100 123 94
112 178 132
o w c
T
$ $
You You
Part Part
100 2 123 4 94 5 1162
112 2 178 4 132 5 1596T C
$2
4
5
O
W
C
C
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One key is to find a way of introducing new material that fits into the jigsaw puzzle created by what is already understood.
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A mouse is placed in the maze above. It is equally likely to pass through any open door. Each time it passes through a doorway is considered a transition.
What is the probability that the mouse will be in Room 4 after 2 transitions if it starts in Room 1?
Transition Matrix
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What is the probability that the mouse will be in Room 4 after 2 transitions if it starts in Room 1?
From 1 to 2 and then from 2 to 4OrFrom 1 to 3 and then from 3 to 4
Can we create a transition matrix for the two-step probailities?
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After 3, 4, 5 and 6 Transitions
After 8, 10, 12 and 100 Transitions
Age Specific Growth
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Begin with [ 100 100 0 0 0 0 0 ] sheep.
How many sheep are in the [2, 4) age group in two years?
How many in the [6, 8) age group in two years?
How many in the [0, 2) age group?
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Photo by John Haslam
After one transition, how many sheep in each group?
After two transitions?
0 100 100 0 0 0 0 0X
1 45 85 97 0 0 0 0X
2 90 38 82 90 0 0 0X http://courses.ncssm.edu/math/talks/conferences/
Matrix Multiplication
1 2 3 4 5 6 7kX P P P P P P P
1 1
2 2
3 3
4 4
5 5
6 6
7
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
B S
B S
B S
L B S
B S
B S
B
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Calculator
1 0
2 1
3 2
1n n
X X L
X X L
X X L
X X L
1 0
22 1 0 0
2 33 2 0 0
1 0n
n
X X L
X X L X L L X L
X X L X L L X L
X X L
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Markov Chains
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Operations with Matrices
Dan TeagueNC School of Science and Mathematics
What is the probability that a taxi from Downtown will be again in Downtown after three fares?
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