GCSEFurther MathsAQARevision Notes5. Matrix TransformationsMatrix TransformationsCombining Matrix Transformations

Combining Matrix Transformations (AQA GCSE Further Maths)

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

Exam Tip

  • 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 must write each transformation as a matrix and use 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.
 

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

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

Simplifying, it can be seen that this is the same as bold A

bold CB bold equals begin bold style stretchy left parenthesis table row cell negative 1 end cell 0 row 0 cell negative 1 end cell end table stretchy right parenthesis end style bold equals bold A

This makes sense geometrically as well: a reflection in the y-axis then the x-axis is equivalent to an enlargement of scale factor -1 (the same as a rotation of 180° about the origin)

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    Mark

    Author: Mark

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.