Matrix

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Matrix
Multiplication
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No, not that Matrix
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Were talking about this kind
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Weve talked about how to add subtract matrices

• To add or subtract matrices, the dimensions of
  each matrix must be the same
• We add or subtract matrices by adding or
  subtracting corresponding elements
• Through this technique, the answer matrix will
  have the same dimensions as those that were added
  or subtracted

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Here are some worked examples
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Now lets talk multiplication

• Using what we know about matrix addition
  subtraction, what would you do to multiply two
  matrices together?

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Is this what you were thinking?

• To multiply matrices, I simply multiply the
  corresponding elements like I did when adding or
  subtracting matrices

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Unfortunately, doing matrix multiplication this
way will give you what I like to call the
WRONG
ANSWER
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• While this is ONE way to multiply matrices,
• this is not the most useful way to do so.
• There is a better way, but as expected, this
• better way takes some work to understand.
• To assist us in our understanding, I have
• called upon my good friend Superman to
• help us out.

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Lets look at matrix multiplication with the
following situation

• Times have been tough
• lately, so Superman has
• been forced to take odd
• jobs to make ends
• meet. Superman aids
• with airport security
• using his x-ray vision.
• He earns 7 an hour
• for his services.

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• Superman uses his strength and flying abilities
• to help stock warehouses.

Superman earns 12 an hour for his services.
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Superman uses his strength and lack
of vulnerability to work as a bouncer for a night
club
Superman earns 15 an hour for his services.
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We can store this informationin a matrix
The row represents the money earned per hour, and
each column represents a different job.
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Now we can create another matrix that shows the
amount of hours that Superman worked at each job
last week.In this matrix each row represents
the hours worked at each job, and the column
represents jobs that Superman works at.
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Now the question we want to ask is
How much money did Superman earn last week?
To find the answer, we will learn how to multiply
matrices!
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Lets set up the problem
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What kind of answer do we expect?

• An answer that is one number
• Being one number, the answer should be a 1 x 1
  matrix

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Is this what you came up with?
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Let's see how that works...
If we think about what the problem means then we
multiply 7 times 12 hours to obtain the money
earned from airport security, 12 times 10 hours
to find the money earned at the warehouse, and
15 times 15 hours to find the money earned
working as a bouncer. Then we ADD these numbers
together to get the total.
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Lets try this again

• Suppose this time that we want to find out
• how much money Superman made every
• week for the past month. Could we use this
• idea of matrix multiplication to get the
• answer?

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Suppose that the matrix below shows the amount of
hours that Superman worked at each job each week
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What kind of answer do we expect?

• Since the question asks how much money Superman
  made for each of the last four weeks, we expect a
  matrix that has four elements

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(No Transcript)
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Now lets look at the dimensions of the matrices
What do you notice about the dimensions?
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Do you suppose that this multiplication will work?
What does your answer confirm about the
dimensions of two matrices in order for them to
be multiplied?
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To sum up

• The dimensions of the answer are the outermost
  dimensions of the multiplicands
• The inner dimensions must be the same in order
  for multiplication to be defined

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Now we know how to multiply matrices!
Whoa!