Revision NotesExam QuestionsFlashcardsPast PapersMock Exams
IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryMatrix TransformationsMatrices of Composite Transformations

Matrices of Composite Transformations (DP IB Applications & Interpretation (AI)): Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

Updated on

DP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HL

Matrices of composite transformations

What is a composite transformation?

  • A composite transformation is the result of applying more than one transformation to an object

    • e.g. a rotation followed by an enlargement

  • The order of transformations is important

    • e.g. a rotation followed by a reflection can be different to the reflection followed by the rotation

  • It is possible to find a single composite function matrix that does the same job as applying the individual transformation matrices

How do I find a single matrix representing a composite transformation?

  • You use matrix multiplication to find the single matrix of a composite transformation

  • The single matrix is bold italic S bold italic T where

    • bold italic T is the matrix for the first transformation

    • bold italic S is the matrix for the second transformation

  • You can extend this to multiple transformations

    • Start with the first matrix

    • Then pre-multiply by the next matrix

    • And so on

Examiner Tips and Tricks

Compare this to composite functions. The function f ring operator g means g is applied first, followed by f.

How do I apply the same transformation matrix more than once?

  • The matrix for the transformation bold italic T applied twice is bold italic T squared

  • The matrix for the transformation bold italic T applied ntimes is bold italic T to the power of n

  • You can use your knowledge of transformations to form the identity matrix

    • e.g. if bold italic R is the matrix for a rotation of 30° clockwise then bold italic R to the power of 12 equals open parentheses table row 1 0 row 0 1 end table close parentheses

    • e.g. if bold italic S is the matrix for a reflection in the line y equals m x then bold italic S squared equals open parentheses table row 1 0 row 0 1 end table close parentheses

Examiner Tips and Tricks

Remember you can use your GDC to do matrix multiplication. You might want to do it by hand first and use your GDC to check.

Worked Example

The matrix E represents an enlargement with scale factor 0.25, centred on the origin. 
The matrix R represents a rotation, 90° anticlockwise about the origin.  

a) Find the matrix, C, that represents a rotation, 90° anticlockwise about the origin followed by an enlargement of scale factor 0.25, centred on the origin.

3-6-1-ib-hl-ai-only-we3a-soltn

b) A square has position matrix bold italic T subscript bold 0 equals open parentheses table row 0 0 256 256 row 0 256 256 0 end table close parentheses.  Tn represents the position matrix of the image square after it has been transformed n times by matrix C.  Find T4

3-6-1-ib-hl-ai-only-we3b-soltn

c) Find the single transformation matrix that would map T4 to T0.

3-6-1-ib-hl-ai-only-we3c-soltn

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Test yourself
    Previous:Matrices of Geometric TransformationsNext:Determinant of a Transformation Matrix
    Author
    Reviewer
    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.