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Operations with Matrices. Dan Teague NC 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. Working with Units.

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Operations with matrices

Operations with Matrices

Dan Teague

NC School of Science and Mathematics

teague@ncssm.edu


Reasoning and sense making
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
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

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


Operations with matrices

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

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


Think about the mathematical practices
Think about the Mathematical Practices traveling in miles per minute?

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Now on to matrices
Now, on to Matrices traveling in miles per minute?

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What is a matrix
What is a Matrix? traveling in miles per minute?

A matrix is a rectangular array of numbers.

A matrix as a model is

a rectangular array of

numbers with meaning.

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


The summer woodworking project
The Summer Woodworking Project traveling in miles per minute?

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
Creating a Matrix traveling in miles per minute?

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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April and may results
April and May Results traveling in miles per minute?

Dimensions of the Matrices

Interpretation of the Elements

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Operations with matrices

How many type II’s did your partner make in the 2 months? traveling in miles per minute?

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?

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


Operations with matrices

http://courses.ncssm.edu/math/talks/conferences/ traveling in miles per minute?


Define this process as addition
Define this process as traveling in miles per minute?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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Operations with matrices

How many of each type did the partners make during June? traveling in miles per minute?

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Now we can define subtraction
Now We Can Define traveling in miles per minute?Subtraction

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Next spring you want to doubled your april output
Next Spring you want to Doubled your April Output traveling in miles per minute?

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Types and wood
Types and Wood traveling in miles per minute?

Interpret the elements in the Wood matrix.

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Operations with matrices

In April, how much Oak did You use? traveling in miles per minute?

In April, how much Cherry did the Partner use?

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Operations with matrices

http://courses.ncssm.edu/math/talks/conferences/ traveling in miles per minute?


Operations with matrices

http://courses.ncssm.edu/math/talks/conferences/ traveling in miles per minute?


Define matrix multiplication
Define Matrix Multiplication traveling in miles per minute?

Just like (ft/mile)∙(miles/hr) = ft/hr

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Cost matrix
Cost Matrix traveling in miles per minute?

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 matrix1
Cost Matrix traveling in miles per minute?

Oak costs 2 cents/strip, Walnut costs 4 cents/strip, Cherry costs 5 cents/strip.

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

Just like (ft/mile)∙(miles/hr) = ft/hr


Operations with matrices

http://courses.ncssm.edu/math/talks/conferences/ traveling in miles per minute?


Operations with matrices

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/


Operations with matrices

A fits into the jigsaw puzzle created by what is already understood.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
Transition Matrix fits into the jigsaw puzzle created by what is already understood.

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Operations with matrices

What is the probability that the mouse will be in Room 4 after 2 transitions if it starts in Room 1?


Operations with matrices

From 1 to 2 after 2 transitions if it starts in Room 1?

and

then from 2 to 4

Or

From 1 to 3

and

then from 3 to 4


Operations with matrices

Can we create a transition matrix for the two-step probailities?

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Age specific growth
Age Specific probailities?Growth

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Operations with matrices

Begin with [ 100 100 0 0 0 0 0 ] sheep. probailities?

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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Operations with matrices

Photo by John Haslam probailities?

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Operations with matrices

After one transition, how many sheep in each group? probailities?

After two transitions?

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Matrix multiplication
Matrix Multiplication probailities?

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Calculator
Calculator probailities?

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Markov chains
Markov Chains probailities?

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Operations with matrices1

Operations with Matrices probailities?

Dan Teague

NC School of Science and Mathematics

teague@ncssm.edu


Operations with matrices

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

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


Operations with matrices

http://courses.ncssm.edu/math/talks/conferences/ again in Downtown after three fares?


Operations with matrices

http://courses.ncssm.edu/math/talks/conferences/ again in Downtown after three fares?