Lecture 9' Matrix Multiplication

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Lecture 9. Matrix Multiplication

• Learning objectives. By the end of this lecture
  you should
• Understand when matrices can be multiplied
• Understand and have practised matrix
  multiplication
• 1. Introduction Multiplication for matrices has
  some special rules.
• The first rule is that the order of
  multiplication matters. In general AxB (or AB) is
  not the same as BA.
• This is very different to multiplying numbers
  where the order doesnt matter e.g. 3x4 4x3
  12.
• Aside
• Addition, multiplication, matrix multiplication
  etc. are examples of operators
• An operator is said to be commutative if x
  operator y y operator x for any x and y
• Addition is commutative xy yx
  multiplication is commutative subtraction is not
  commutative (2-1 ? 1-2).
• Matrix multiplication is also not commutative.

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Multiplying two matrices.

• The second rule is that you can only multiply two
  matrices if they are conformable.
• Two matrices are conformable if the number of
  columns for the first matrix is the same as the
  number of rows for the second matrix.
• If the matrices are not conformable they cannot
  be multiplied.
• Example 1 does AB exist?
• Answer A is a 2x4 matrix. B is a 3x3. So A has 4
  columns and B has 3 rows. Therefore AB does not
  exist. A and B are not conformable.

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Multiplying two matrices.

• Example 2 does AB exist?
• Answer A is a 2x5 matrix. B is a 5x2. So A has 5
  columns and B has 5 rows. A and B are
  conformable. Therefore AB exists.
• Finding AB.
• Suppose A and B are conformable. So C AB
  exists. To calculate it
• To get the first element on the first row of C
  take the first row of A and multiply each element
  in turn against its corresponding element in the
  first column of B. Add the result. Well call
  this multiplying the first row of A against the
  first column of B.
• Example c11 4x4 -1x-1 3x3 3x3 0x1 35
• To get the remaining elements in the first row
  repeat this procedure with the first row of A
  multiplying each column of B in turn.

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Multiplying two matrices.
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Multiplying two matrices.
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Multiplying two matrices - formally.

• Suppose A is an mxn matrix and and B is an nxr
  matrix with typical elements aik and bkj
  respectively
• Then AB C where element cij is
• Note that the result is an mxr matrix

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Another example.

• Example 2 calculate AB
• First we note that A is 1x4 and B is 4x2, so AB
  exists and is a 1x2 matrix.
• The first element c11 (1)(1)(0)(2)(2)(1)(0)(0
  ) 3
• The second element c12 (1)(0)(0)(0)(2)(3)(0)(
  1) 6
• So C (3 6)

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Quiz.

• Can you multiply the following matrices? If so,
  what is the dimension of the result?
• BA
• BC
• AAT
• ATA
• CC

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More multiplication.

• We saw from the quiz that the order of
  multiplication matters. ATA had different
  dimensions to AAT, while BA did not even exist.
• Remember that the first row of AB is found by
  multiplying the first row of A by the columns of
  B in turn.
• The second row comes from multiplying the second
  row of A by the columns of B in turn
• And so on.
• Finding CC (sometimes written C2).
• You find AAT

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Multiplying square matrices by the identity matrix

• Recall
• A square matrix has the same number of rows and
  columns its nxn
• The identity matrix is a square matrix with 1s in
  the leading diagonal and 0s everywhere else. E.g.
• Finding IA then find AI when.

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A general result for square matrices

• If A is a square matrix then IA AI A.
• You can prove AI A.
• NB. This only applies to square matrices

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Inverses for square matrices

• Idea
• In normal multiplication every number except zero
  has an inverse.
• The inverse of 3 is 1/3 the inverse of 27 is
  1/27, the inverse of -1.1 -1/1.1
• A number times its inverse equals 1 x(1/x) 1
• And the inverse times the number equals 1 (1/x)x
  1
• The inverse of the inverse is the original
  number.
• We define an inverse of a square matrix in a
  similar manner.
• If A is an nxn matrix then the inverse of A,
  written A-1 is an nxn matrix such that
• AA-1 I
• A-1A I
• Notes
• This means that A is the inverse of A-1
• I is its own inverse.
• A-1 may not exist.

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Inverse matrix example

• Suppose and
• Then
• What is BA?

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Quiz II.

• Find BC
• Is D the inverse of C

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Summary.

• 4 Definitions you should now memorise
• conformable, commutative, matrix multiplication,
  inverse
• 3 skills you should be able to do
• Check whether two matrices are conformable
• Multiply two conformable matrices
• Show whether A and B are inverses.
• Next determinants and finding the inverse