Mathematical Operations on Matrices 1

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Mathematical Operations on Matrices 1
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What you need to know

• General algebraic methods
• Use of functions
• Some knowledge of vectors and how to manipulate
  them
• The content of Introduction to Matrices.
• IMPORTANT NOTE MAKE SURE YOU ARE COMFOTABLE WITH
  THE ABOVE BEFORE PROCEEDING WITH THIS SECTION.

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MATHS AND MATRICES

• This section will cover some simple mathematical
  operations using matrices
• Addition (covered in part 1)
• Subtraction (covered in part 1)
• Multiplication by a scalar (covered in part 2)
• Multiplication by another matrix (covered in part
  2)
• The Inverse Matrix (covered in part 3)
• It will also give practical examples and
  exercises in representing and manipulating data
  in a matrix.
• As you may have noticed, this topic has been
  split into three separate presentations, entitled
  Mathematical Methods and numbered 1 to 3.
  These should be worked through in order.

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Storing Information

• As you will remember, a matrix is a rectangular
  array of numbers, and can be a useful way of
  storing information.
• Hopefully you remember matrix A
• 1 3 5 7 A 2 4 6
  8 5 9 0 2
• In this form it could be storing any information.
  In this respect the matrix could be interpreted
  differently depending on the situation. The
  structure does not decide the content, and this
  should be borne in mind as we use this matrix.

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Storing Information 2

• Previously, we considered that the matrix was an
  example of sales data from a hardware store. We
  decided that it showed the sales of three
  products over the course of four days. A was
  considered as a form of table, in which we have
  stored the following information Mon
  Tues Wed Thurs PANS 1 3 5
  7 NAILS 2 4 6 8 SAWS 5
  9 0 2

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Addition using Matrices

• Suppose that we had a companion store in another
  town, such that one is in town 1 and the other in
  town 2, and we would like to be able to calculate
  the total sales of the items in both outlets.
• We create a matrix of the data from the second
  shop in a matrix of the same kind as the first.
  This gives us matrix B.
• 1 3 5 7 2 4 4 9 A 2 4
  6 8 B 3 3 7 7 5 9 0 2 8 6
  4 1
• To add two matrices of the same kind, we can
  simply add the corresponding elements, as we
  shall see.

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Addition with Matrices 2

• 1 3 5 7 2 4 4 9 A B 2
  4 6 8 3 3 7 7 5 9 0 2
  8 6 4 1
• 12 34 54 79 A B 23
  43 67 87 58 96 04 21
• 3 7 9 16 A B 5 7 13
  15 C 13 15 4 3
• Can you understand how this works?

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Summary of Addition with Matrices

• If two matrices are of the same kind, they can be
  added together to produce a new matrix of the
  same kind.
• This means that if Aai j and Bbi j are both
  of the same kind the following can be presumed
• C A B ai j bi j
• It should also be noted that matrices have
  commutative properties. This means that in the
  case of matrix addition A B B A. In
  simple terms, this means that for addition, the
  order of manipulation, or the order of the
  matrices, does not affect the result (as long as
  they are added correctly!).

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Subtraction of Matrices

• Subtraction of matrices works much the same as
  addition. Suppose we want to find out whether
  the first shop, represented by A, sold more of
  the items than shop B.
• 1 3 5 7 2 4 4 9 A 2 4
  6 8 B 3 3 7 7 5 9 0 2 8 6
  4 1
• How would you do this? Try to work it out and
  then go to the next page for a worked example.

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Subtraction of Matrices 2

• 1 3 5 7 2 4 4 9 A- B 2
  4 6 8 - 3 3 7 7 5 9 0 2
  8 6 4 1
• 1-2 3-4 5-4 7-9 A -B 2-3
  4-3 6-7 8-7 5-8 9-6 0-4 2-1
• -1 -1 1 -2 A -B -1 1
  -1 1 D -3 3 -4 1
• Can you see how this works? In this case A does
  not seem to have sold much more than B, and may
  have possibly sold less on average.

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Subtraction of Matrices 3

• The best way to find the differences between two
  matrices of the same kind is actually using
  addition, by using the Opposite Matrix of the
  matrix we want to subtract with A B
  A -B -B A
• We calculate the opposite matrix, by changing the
  signs, and then add that to the matrix we want to
  subtract from.
• NB. It may seem unnecessarily long-winded, but it
  is less prone to mistakes, and easier to check
  for errors.

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Exercise 2 See Answers.ppt

• Add the following matrices, where possible
• 5 7 6 8
• 10 12 8 12 15 17
  9 42 7 15 4
  13
• 9 8 7 6 5 4 4 5 6
  7 8 9 1 3 2

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Exercise 2 Continued

• Subtract the following matrices, where possible
• 5 - 7 6 8
• 10 12 - 8 12 15 17
  9 42 7 15 4 13
• 9 8 7 - 6 5 4 4 5 6
  7 8 9 1 3 2
• -9 8 7 - -6 5 4 4 -5 6
  7 -8 -9 1 -3 2

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Summary

• In this section, you have learnt how to
  manipulate matrices to perform the simple
  mathematical operations of addition and
  subtraction.
• Please make sure that you are clear on the topics
  that have been covered in this part before
  proceeding to the next part of this course
  Mathematical Operations on Matrices 2, where
  you will be shown how to multiply a matrix by a
  scalar, and how to multiply two matrices
  together.
• Remember that the answers to the set questions
  can be found in Answers.ppt.