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Enlargement Matrices (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Enlargement matrices

How do I find enlargement matrices?

  • Imagine the unit square OABC

    • It has a side-length of 1 unit

    • O is the origin

unit-square

  • The coordinates of A and C as column vectors are

    • A equals open parentheses table row 1 row 0 end table close parentheses and C equals open parentheses table row 0 row 1 end table close parentheses

  • Under an enlargement of scale factor k with centre at the origin (including negative scale factors), A moves to A' and C moves to C' 

    • The matrix, M representing this enlargement is bold M equals open parentheses table row cell A apostrophe space vertical line end cell cell C apostrophe end cell end table close parentheses

    • A' and C' are column vectors of the new positions

      • So bold M is a 2×2 matrix

    • The points O and B are not needed, as we can draw the enlarged square using just A' and C' (as O won't move)

  • A apostrophe equals open parentheses table row k row 0 end table close parentheses and C apostrophe equals open parentheses table row 0 row k end table close parentheses

    • They are both just moving along the x and y axes respectively

  • So all enlargement matrices have the form bold M equals open parentheses table row k 0 row 0 k end table close parentheses

    • This is the same as bold M equals k bold I, where bold I is the identity matrix

  • For example:

    • The matrix representation of an enlargement of scale factor 3 with centre at the origin is open parentheses table row 3 0 row 0 3 end table close parentheses

    • The matrix representation of an enlargement of scale factor negative 1 half with centre at the origin is open parentheses table row cell negative 1 half end cell 0 row 0 cell negative 1 half end cell end table close parentheses

Worked Example

The matrix M representing a transformation is given by open parentheses table row cell 1 fourth end cell 0 row 0 cell 1 fourth end cell end table close parentheses.

Describe geometrically the transformation represented by M.

Answer:
  
The matrix bold M can be written as a multiple of the identity matrix, bold I

open parentheses table row cell 1 fourth end cell 0 row 0 cell 1 fourth end cell end table close parentheses equals 1 fourth open parentheses table row 1 0 row 0 1 end table close parentheses

So the unit square is being scaled by 1 fourth

Enlargement by scale factor 1 fourthwith centre at the origin

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    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.