operations with matrices dan teague nc school of science and mathematics teague@ncssm.edu

Operations with Matrices

Dan TeagueNC School of Science and Mathematics

teague@ncssm.edu

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

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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

teague@ncssm.edu

What is the probability that a taxi from Downtown will be again in Downtown after three fares?

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