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A LevelFurther Maths: Core PureEdexcelRevision Notes2. Matrices2.2 Transformations using Matrices2.2.1 Transformations using a Matrix

Transformations using a Matrix (Edexcel A Level Further Maths: Core Pure)

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Paul

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Paul

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Transformations using a Matrix

What is a transformation matrix?

  • A transformation matrix is used to determine the coordinates of an image from the transformation of an object
    • reflections, rotations, enlargements and stretches
    • Commonly used transformation matrices include
  • (In 2D) a multiplication by any 2x2 matrix could be considered a transformation (in the 2D plane)
    • This can be done similarly in higher dimensions
  • An individual point in the plane can be represented as a position vector, open parentheses table row x row y end table close parentheses
    • Several points, that create a shape say, can be written as a position matrix space open parentheses table row cell x subscript 1 end cell cell x subscript 2 end cell cell x subscript 3 end cell cell... end cell row cell y subscript 1 end cell cell y subscript 2 end cell cell y subscript 3 end cell cell... end cell end table close parentheses
  • A matrix transformation will be of the form open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses
    • where open parentheses table row x row y end table close parentheses represents any point in the 2D plane
    •  open parentheses table row a b row c d end table close parentheses is a given matrix

How do I find the coordinates of an image under a transformation?

  • The coordinates (x’, y’) - the image of the point (x, y) under the transformation with matrix open parentheses table row a b row c d end table close parentheses are given by

open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses

  • Similarly, for a position matrix

open parentheses table row cell x apostrophe subscript 1 end cell cell x apostrophe subscript 2 end cell cell x apostrophe subscript 3 end cell cell... end cell row cell y apostrophe subscript 1 end cell cell y apostrophe subscript 2 end cell cell y apostrophe subscript 3 end cell cell... end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row cell x subscript 1 end cell cell x subscript 2 end cell cell x subscript 3 end cell cell... end cell row cell y subscript 1 end cell cell y subscript 2 end cell cell y subscript 3 end cell cell... end cell end table close parentheses 

  • A calculator can be used for matrix multiplication
    • If matrices involved are small, it may be as quick to do this manually 

  • STEP 1
    Determine the transformation matrix (T) and the position matrix (P)
    The transformation matrix, if uncommon, will be given in the question
    The position matrix is determined from the coordinates involved, it is best to have the coordinates in order, to avoid confusion 

  • STEP 2
    Set up and perform the matrix multiplication required to determine the image position matrix, P’
    P’     = TP 

  • STEP 3
    Determine the coordinates of the image from the image position matrix, P’

How do I find the coordinates of the original point given the image under a transformation?

  •  To ‘reverse’ a transformation we would need the inverse transformation matrix
    • i.e. T-1
    • For a 2x2 matrix open parentheses table row a b row c d end table close parentheses the inverse is given by fraction numerator 1 over denominator det bold italic T end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses
      • where det bold italic T equals a d minus b c
    • A calculator can be used to work out inverse matrices
  • You would rearrange open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses
    • fraction numerator 1 over denominator det space bold italic T end fraction stretchy left parenthesis table row d cell negative b end cell row cell negative c end cell a end table stretchy right parenthesis stretchy left parenthesis table row cell x apostrophe end cell row cell y apostrophe end cell end table stretchy right parenthesis equals stretchy left parenthesis table row x row y end table stretchy right parenthesis

Examiner Tip

  • Read the question carefully to determine if you have the points before or after a transformation

Worked example

A quadrilateral, Q, has the four vertices A(2, 5), B(5, 9), C(11, 9) and D(8, 5).

Find the coordinates of the image of Q under the transformation bold italic T equals open parentheses table row 3 cell negative 1 end cell row cell negative 1 end cell 2 end table close parentheses.

2-2-1-edx-a-fm-we1-soltn

Determinant of a Transformation Matrix

What does the determinant of a transformation matrix (A) represent?

  •  The absolute value of the determinant of a transformation matrix is the area scale factor (2D) or volume scale factor (3D)
    • Area scale factor = |det A| if 2x2
    • Volume scale factor = |det A| if 3x3
  • The area/volume of the image will be product of the area/volume of the object and |det A|
    • Area of image = |det A| × Area of object (if 2x2)
    • Volume of image = |det A| × Volume of object (if 3x3)
  • Note the area will reduce if |det A| < 1
  • If the determinant is negative then the orientation of the shape will be reversed
    • For example: the shape has been reflected

How do I solve problems involving the determinant of a transformation matrix?

  • Problems may involve comparing areas of objects and images
    • This could be as a percentage, proportion, etc
  • Missing value(s) from the transformation matrix (and elsewhere) can be deduced if the determinant of the transformation matrix is known
  • Remember to use the absolute value of the determinant
    • This can lead to multiple answers to equations
    • Use your calculator to solve these

Worked example

An isosceles triangle has vertices A(3, 1), B(15, 1) and C(9, 9).

a)
Find the area of the isosceles triangle.

2-2-1-edx-a-fm-we2a-soltn

b)
Triangle △ABC is transformed using the matrix bold italic T equals open parentheses table row 3 2 row cell negative 1 end cell 2 end table close parentheses. Find the area of the transformed triangle.

2-2-1-edx-a-fm-we2b-soltn

c)
Triangle △ABC is now transformed using the matrix bold italic U equals open parentheses table row a cell negative 2 end cell row 3 cell a squared end cell end table close parentheses where a element of straight integer numbers. Given that the area of the image is twice as large as the area of the object, find the value of a.

2-2-1-edx-a-fm-we2c-soltn

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.