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Reflection Matrices (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Reflection matrices

How do I find reflection matrices?

  • Imagine the unit square OABC

    • It has a side-length 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 a reflection about an axis (or y = ± x), A moves to A' and C moves to C' 

    • The matrix, M representing this reflection 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 reflected square using just A' and C' (as O won't move)

  • For example:

    • To find the matrix representing a reflection about the x-axis

      • A stays where it is, so A apostrophe equals open parentheses table row 1 row 0 end table close parentheses

      • C goes to C apostrophe equals open parentheses table row 0 row cell negative 1 end cell end table close parentheses (on the negative y-axis)

      • bold M equals open parentheses table row cell A apostrophe space vertical line end cell cell C apostrophe end cell end table close parentheses equals open parentheses table row 1 0 row 0 cell negative 1 end cell end table close parentheses

    • To find the matrix representing a reflection in the line y = x

      • A goes to A apostrophe equals open parentheses table row 0 row 1 end table close parentheses (on the positive y-axis)

      •  C goes to C apostrophe equals open parentheses table row 1 row 0 end table close parentheses (on the positive x-axis)

      • bold M equals open parentheses table row cell A apostrophe space vertical line end cell cell C apostrophe end cell end table close parentheses equals open parentheses table row 0 1 row 1 0 end table close parentheses

        • This is not the same as the identity matrix as the 1s are on the wrong diagonal

Worked Example

(a) The matrix M represents a reflection in the y-axis. Work out M.

Answer:

Consider how the points A and C on the unit square are transformed by a reflection in the y-axis

TuW80_W4_reflection-matrix-we-1

The point A open parentheses table row 1 row 0 end table close parentheses moves to A' open parentheses table row cell negative 1 end cell row 0 end table close parentheses 

The point C open parentheses table row 0 row 1 end table close parentheses remains in the same place

The transformation matrix is given by bold M equals open parentheses table row cell A apostrophe space vertical line end cell cell C apostrophe end cell end table close parentheses 

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

(b) The matrix N represents a reflection in the line y equals negative x. Work out N.

Answer:

Consider how the points A and C on the unit square are transformed by a reflection in the line y equals negative x

reflection-matrix-we-2

The point A open parentheses table row 1 row 0 end table close parentheses moves to A' open parentheses table row 0 row cell negative 1 end cell end table close parentheses 

The point C open parentheses table row 0 row 1 end table close parentheses moves to C' open parentheses table row cell negative 1 end cell row 0 end table close parentheses

The transformation matrix is given by bold N equals open parentheses table row cell A apostrophe space vertical line end cell cell C apostrophe end cell end table close parentheses 

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

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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.