Further Pure 1

1
Further Pure 1

• Inverse Matrices

2
Reminder from lesson 1

• Note that any matrix multiplied by the identity
  matrix is itself.
• And any matrix multiplied by the zero matrix is
  the zero matrix.

3
Inverse Matrix

• All operations have an opposite.
• We discussed in lesson 2 about using matrices to
  perform transformations.
• An inverse matrix will undo the transformation
  and return you to where you started.
• If a matrix is called A, then its inverse is
  known as A-1.
• In lesson 1 we briefly met the concept of an
  identity matrix (seen on first slide).
• So if multiplying A by A-1 returns you to where
  you started and multiplying by the identity
  matrix leaves you where you are, we can conclude
  that AA-1 A-1A I

4
Challenge 1

• You need to know the general formula for the
  inverse of a 2 2 matrix.
• Can you find the inverse to the following matrix?
• Use the property TT-1 I
• From this we can form 2 pairs of simultaneous
  equations.
• 2p 3q 1 2r 3s 0
• p 4q 0 r 4s 1

5
Challenge 1

• The solutions to these equations are
• So the inverse of T is

6
Challenge 1

• Lets now take any 2 2 matrix.
• Can you use T and T-1 to find the inverse of M
  and hence the general formula for the inverse of
  any 2 2 matrix?

7
The Determinant of a 2 2 matrix

• We have just found the general equation for the
  inverse of any 2 2 matrix.
• The ? symbol is a capital delta and will always
  be a numerical value.
• The value can be calculated from the matrix and
  is known as the determinant of the matrix.
• Using T and T-1 can you spot how to calculate it?

8
The Determinant of a 2 2 matrix

• To calculate the determinant of a matrix M you
  multiply a by d and subtract b by c.
• Below is the official notation.
• If the det is zero then the inverse does not
  exist and the matrix is known as singular.
• If the det is not zero then the inverse does
  exist and the matrix is known as non-singular.
• Note Only square matrices have inverses.

9
Challenge 2

• This is above the scope of the course and not
  required for you to do.
• However it is a challenging question that will
  test your algebraic manipulation skills.
• Can you find the inverse of M using the identity
  below and the method we used a few slides ago.
• This will also prove where the formula for the
  determinant comes from.

10
Questions

• Find the inverse of the following matrices.

11
Inverse of a product

• Find the inverse of AB.
• Lets call the inverse of AB, X.
• So as we already know X(AB) I
• First multiply by B-1 X(AB)B-1 IB-1
• XA B-1
• Next multiply by A-1 XAA-1 B-1A-1
• X B-1A-1
• This is an important result that you need to know
• (AB)-1 B-1A-1

12
Properties of the determinant

• The orange square is an enlargement of the black
  square by a scale factor 2.
• What is the area of the object?
• Area 9 units2
• What is the area of the image?

• Area 36 units2
• The transformation performed can be described by
  the following matrix
• What do you notice about the determinant of the
  matrix and the enlargement shown.
• The determinant of a matrix indicates the scale
  factor of the area of enlargement.
• The det T is known as the signed scale factor as
  it can be negative.
• The negative signifies that the rotation
  direction has been reversed.

13
Task

• Can you explain how we know that the area of any
  shape rotated ? degrees anti-clockwise about the
  origin remains the same.

14
Matrices with det 0

• The determinant of a matrix tells us the scale
  factor of the areas enlargement.
• What would be the area of a shape transformed by
  a matrix with det 0?
• The area would be 0.
• All the points will have been transformed so what
  will the image look like?
• The image will be a straight line.
• We can see an example of this on the next slide.

15
Example 1

• Lets start with a rectangle on a 2D pair of axes.
• We can write the co-ordinates of the vertices in
  matrix form.
• Next transform the object using a matrix with a
  det 0
• The image becomes a series of points that are in
  a straight line.

16
Example 1

• In fact although we used a rectangle for the
  example any point in the plane will transform to
  the line.
• From the diagram its clear to see what the
  equation of the line will be.
• y x

17
Example 1

• We can reach the same result as the last slide
  using an algebraic method.
• Lets look at the general co-ordinate (x,y).
• Under the transformation we get the co-ordinates
  (x,y)
• Using matrix multiplication we can see that.
• x 2y x
• x 2y y
• From this we get y x 2y x
• Or y x

18
Example 2

• The plane is transformed by the
  matrix.
• Show that the whole plane is mapped to a straight
  line and find the equation of this line.
• Using matrix multiplication gives us the
  simultaneous equations.
• x 2x y
• y -4x 2y
• From the equations we get y -2(2x y) -2x
• All the points will map to the line y -2x

19
Example 2

• All points in a plane transform to a straight
  line.
• This is because there are infinitely many lines
  that transform to a single point.

20
Example 3

• For the matrix T find the equation of the line of
  points that map to (5,-10).
• We use matrix multiplication to find what
  equations will be equal to the co-ordinate
  (5,-10)
• This gives us the equations
• 2x y 5
• -4x y -10
• These two equations give the exact same
  information.
• 2x y 5

21
Summary 1

• The inverse of a matrix
• Is
• Where

22
Summary 2

• MM-1 M-1M I
• X B-1A-1