IGCSEMathsEdexcelMaths BRevision NotesMatricesTransformations using MatricesCombining Matrix Transformations

Combining Matrix Transformations (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Combining transformation matrices

How do I find a single matrix that represents a combination of transformations?

  • A point (x, y) can be transformed twice

    • First by the matrix bold P, then second by the matrix bold Q

    • This is called a combined (or composite) transformation

  • A single matrix, bold M, representing the combined transformation can be found using matrix multiplication as follows: 

    • bold M equals bold QP 

      • The order matters: the first transformation is the last in the multiplication

      • The order is the reverse of what you may expect! 

        • The first transformation is on the right, with the second transformation to its left

    • bold PQ would represent bold Q first, followed by bold P 

Examiner Tips and Tricks

If a question asks you to prove a geometric fact about combined transformations "using matrix multiplication", you cannot just draw a sequence of diagrams for your answer

  • You should write each transformation as a matrix

  • and combine them using QP or PQ (depending on the order)

Worked Example

Three transformations in the x-y plane are given below.

bold A equals open parentheses table row cell negative 1 end cell 0 row 0 cell negative 1 end cell end table close parentheses  represents an enlargement by scale factor -1 about the origin
bold B equals open parentheses table row cell negative 1 end cell 0 row 0 1 end table close parentheses represents a reflection in the y-axis
bold C equals open parentheses table row 1 0 row 0 cell negative 1 end cell end table close parentheses represents a reflection in the x-axis

Use matrix multiplication to prove that A is the same as B followed by C.

Answer:

Transformation bold B followed by transformation bold C would be combined into a single matrix by finding bold CB (note the order)

Find the matrix multiplication bold CB

table row cell bold CB space end cell equals cell space open parentheses table row 1 0 row 0 cell negative 1 end cell end table close parentheses cross times open parentheses table row cell negative 1 end cell 0 row 0 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 cross times open parentheses negative 1 close parentheses plus 0 cross times 0 end cell cell 1 cross times 0 plus 0 cross times 1 end cell row cell 0 cross times open parentheses negative 1 close parentheses plus open parentheses negative 1 close parentheses cross times 0 end cell cell 0 cross times 0 plus open parentheses negative 1 close parentheses cross times 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 plus 0 end cell cell 0 plus 0 end cell row cell 0 plus 0 end cell cell 0 minus 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 0 row 0 cell negative 1 end cell end table close parentheses end cell end table

This is the same as bold A

  • This makes sense geometrically as well

    • A reflection in the y-axis followed by a reflection in the x-axis

    • is equivalent to an enlargement of scale factor -1 (which is the same as a rotation of 180° about the origin)

CB equals stretchy left parenthesis table row cell negative 1 end cell 0 row 0 cell negative 1 end cell end table stretchy right parenthesis equals straight A

Unlock more, it's free!

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

the (exam) results speak for themselves:

    Previous:Determinant of a Transformation MatrixNext:Basic Angle Properties
    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Jamie Wood

    Reviewer: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.