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Standard Matrix Transformations (Edexcel International AS Further Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

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

The 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 equals plus-or-minus x, A moves to A' and C moves to C'  

    • The matrix, bold 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 their new positions

      • The points O  and B  are not needed, as we can draw the reflected square using just A' and C' (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 equals 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 bold M represents a reflection in the y-axis.

Work out bold M.

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 

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

(b) Describe fully the transformation represented by the matrix bold N equals open parentheses table row 0 cell negative 1 end cell row cell negative 1 end cell 0 end table close parentheses.

 

Consider how the points A and C on the unit square are transformed

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

It helps to draw a picture of the unit square being transformed with vertices clearly labelled

reflection-matrix-we-2

This transformation could be a rotation of 180° about O or a reflection in y equals negative x
The vertices A' and C' are in the correct places for a reflection, but not a rotation

The matrix N represents a reflection in the line y equals negative x

Enlargement & Stretch Matrices

Which matrix represents an enlargement?

  • The matrix bold M equals open parentheses table row k 0 row 0 k end table close parentheses represents an enlargement of scale factor k about the origin, O

    • This is the same as bold M equals k bold I

      • bold I is the identity matrix

Which matrix represents a stretch?

  • The matrix bold M equals open parentheses table row a 0 row 0 1 end table close parentheses represents a stretch parallel to the x-axis of scale factor a

    • The point open parentheses x comma space y close parentheses becomes open parentheses a x comma space y close parentheses

  • The matrix bold M equals open parentheses table row 1 0 row 0 b end table close parentheses represents a stretch parallel to the y-axis of scale factor b

    • The point open parentheses x comma space y close parentheses becomes open parentheses x comma space b y close parentheses

  • The matrix bold M equals open parentheses table row a 0 row 0 b end table close parentheses represents a combined stretch of scale factor a parallel to the x-axis and scale factor b parallel to the y-axis

    • If a equals b, the combined stretch is an enlargement

Examiner Tips and Tricks

Use phrases like "parallel to the x-axis" or "parallel to the y-axis" to describe stretches (not "left" or "up"!)

Worked Example

A transformation is represented by the matrix bold M equals open parentheses table row cell 3 plus p end cell 0 row 0 cell 3 minus p end cell end table close parentheses.

Describe fully the transformation in each of the following cases:

(a) p equals 0

Substitute in p equals 0

bold M equals open parentheses table row cell 3 plus 0 end cell 0 row 0 cell 3 minus 0 end cell end table close parentheses equals open parentheses table row 3 0 row 0 3 end table close parentheses

This has the form bold M equals open parentheses table row k 0 row 0 k end table close parentheses where k equals 3

bold M represents an enlargement of scale factor 3 about the origin

You must give its scale factor and centre of enlargement

(b) p equals 2

Substitute in p equals 2