Operations with Matrices Dan Teague NC School of Science and Mathematics [email protected].

Operations with Matrices

Dan TeagueNC School of Science and Mathematics

[email protected]

mailto:[email protected]

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.

http://courses.ncssm.edu/math/talks/conferences/

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


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


miles minutes2 60 ?hour hour

miles 1 hours 1 miles2hour 60 minute 30 minute

Think about the Mathematical Practices


Now, on to Matrices


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.


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


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.


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


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


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


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.


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


Now We Can Define Subtraction


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


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


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


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


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


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?


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

http://courses.ncssm.edu/math/talks/conferences/Just like (ft/mile)∙(miles/hr) = ft/hr

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


One key is to find a way of introducing new material that fits into the jigsaw puzzle created by what is already understood.


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


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?


After 3, 4, 5 and 6 Transitions

After 8, 10, 12 and 100 Transitions

Age Specific Growth


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?



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


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


Markov Chains


Operations with Matrices

Dan TeagueNC School of Science and Mathematics

[email protected]

mailto:[email protected]

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


Operations with Matrices Dan Teague NC School of Science and Mathematics [email protected].

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