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Matrix Transformations (DP IB Applications & Interpretation (AI)): Revision Note

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Transformation by 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

    • Commonly used transformation matrices include

      • reflections, rotations, enlargements and stretches

  • (In 2D) a multiplication by any 2x2 matrix could be considered a transformation (in the 2D plane)

  • 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 plus open parentheses table row e row f 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 and open parentheses table row e row f end table close parentheses are given matrices

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 matrices open parentheses table row a b row c d end table close parentheses and open parentheses table row e row f 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 plus open parentheses table row e row f 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 x subscript 1 end cell cell x subscript 2 end cell cell x subscript 3 end cell cell... end cell end table close parentheses plus open parentheses table row e e e cell... end cell row f f f cell... end cell end table close parentheses

    • If you use this method then remember to add e and f to each column

  • A GDC 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 and addition 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 GDC 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 plus open parentheses table row e row f end table close parentheses

Examiner Tips and Tricks

  • The formula for the determinant and inverse of a 2x2 matrix can be found in the Topic 1: Number and Algebra section of the formula booklet

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.

3-6-1-ib-hl-ai-only-we1a-soltn

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    Paul

    Author: Paul

    Expertise: Maths Content Creator

    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.