International A LevelFurther Maths: Further Pure 1EdexcelRevision NotesMatricesTransformations using MatricesStandard Matrix Transformations

Standard Matrix Transformations (Edexcel International A Level Further Maths)

Revision Note

Mark Curtis

Expertise

Maths

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

Exam Tip

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

bold M equals open parentheses table row cell 3 plus 2 end cell 0 row 0 cell 3 minus 2 end cell end table close parentheses equals open parentheses table row 5 0 row 0 1 end table close parentheses

This has the form bold M equals open parentheses table row a 0 row 0 1 end table close parentheses where a equals 5

bold M represents a stretch of scale factor 5 parallel to the x-axis

You must give its scale factor and direction

Rotation Matrices

How do I find matrices for rotations by multiples of 90°?

  • 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 a rotation about the origin, A moves to A' and C moves to C' 

    • The matrix, bold M representing this rotation 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 rotated square using just A' and C' (O won't move)

  • For example:

    • To find the matrix representing a rotation of 90° anticlockwise about the origin

      • 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 cell negative 1 end cell row 0 end table close parentheses (on the negative 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 cell negative 1 end cell row 1 0 end table close parentheses

    • To find the matrix representing a rotation of 180° about the origin

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

      • 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 cell negative 1 end cell 0 row 0 cell negative 1 end cell end table close parentheses

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

Exam Tip

Students often confuse rotations of 180° with reflections in the lines y equals plus-or-minus x .

How do I find matrices for rotations by any angle?

  • A rotation anticlockwise by any angle, theta, about the origin is represented by the matrix:

    • bold M equals open parentheses table row cell cos space theta end cell cell negative sin space theta end cell row cell sin space theta end cell cell cos space theta end cell end table close parentheses

      • theta can be in degrees or radians

  • A negative value of theta represents a clockwise rotation

    • Remember that sin open parentheses negative theta close parentheses equals negative sin space theta but that cos open parentheses negative theta close parentheses equals cos space theta

  • You can substitute in multiples of 90° to get the matrices above

  • You may be required to recognise common angles from their ratios

    • For example, fraction numerator square root of 3 over denominator 2 end fraction equals cos open parentheses straight pi over 6 close parentheses

Exam Tip

You are given the rotation matrix in the Formulae Booklet.

Worked Example

(a) Describe fully the transformation represented by the matrix bold P equals open parentheses table row 0 1 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 1 row 0 end table close parentheses

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

A rotation of 90 degrees anticlockwise

This transformation could be a rotation of 90° clockwise about O or a reflection in the x-axis
The vertices A' and C' are not in the correct places for a reflection, but are for a rotation

The matrix P represents a rotation of 90° clockwise about the origin

You must give its angle, direction and centre of rotation
270° anticlockwise would also be accepted

(b) Find bold Q, the matrix that represents a clockwise rotation of 120° about the origin, giving your answer in an exact form.

The matrix for an anticlockwise rotation by theta isopen parentheses table row cell cos space theta end cell cell negative sin space theta end cell row cell sin space theta end cell cell cos space theta end cell end table close parentheses
theta is negative, as the rotation is clockwise

theta equals negative 120 degree

Substitute this value of theta into the matrix
Use that cos open parentheses negative theta close parentheses equals cos space theta and that sin open parentheses negative theta close parentheses equals negative sin space theta

open parentheses table row cell cos open parentheses negative 120 degree close parentheses end cell cell negative sin open parentheses negative 120 degree close parentheses end cell row cell sin open parentheses negative 120 degree close parentheses end cell cell cos open parentheses negative 120 degree close parentheses end cell end table close parentheses equals open parentheses table row cell cos open parentheses 120 degree close parentheses end cell cell sin open parentheses 120 degree close parentheses end cell row cell negative sin open parentheses 120 degree close parentheses end cell cell cos open parentheses 120 degree close parentheses end cell end table close parentheses

Use a calculator to find these values (or common angles and symmetry)

bold Q equals stretchy left parenthesis table row cell negative 1 half end cell cell fraction numerator square root of 3 over denominator 2 end fraction end cell row cell negative fraction numerator square root of 3 over denominator 2 end fraction end cell cell negative 1 half end cell end table stretchy right parenthesis

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    Mark Curtis

    Author: Mark Curtis

    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.