Operations with Matrices

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# Operations with Matrices - PowerPoint PPT Presentation

Operations with Matrices. Dan Teague NC School of Science and Mathematics [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. Working with Units.

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### Operations with Matrices

Dan Teague

NC School of Science and Mathematics

[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.

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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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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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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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April and May Results

Dimensions of the Matrices

Interpretation of the Elements

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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?

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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?

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Now We Can Define Subtraction

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Next Spring you want to Doubled your April Output

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Types and Wood

Interpret the elements in the Wood matrix.

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In April, how much Oak did You use?

In April, how much Cherry did the Partner use?

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Define Matrix Multiplication

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

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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.

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

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 4

Or

From 1 to 3

and

then from 3 to 4

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

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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

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After one transition, how many sheep in each group?

After two transitions?

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Matrix Multiplication

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Calculator

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Markov Chains

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### Operations with Matrices

Dan Teague

NC School of Science and Mathematics

[email protected]

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

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