GCSEFurther MathsAQARevision Notes5. Matrix TransformationsMatrix TransformationsTransformations with Matrices

Transformations with Matrices (AQA GCSE Further Maths)

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Transforming a Point

How do I transform a point using a matrix?

  • A point (x, y) in a 2D plane can be transformed on to another point (x',y') by a matrix, M
    • (x, y) is the object and (x',y') is the image
  • The coordinates of the image point can be found using matrix multiplication
  • To transform (x, y) by the matrix open parentheses table row a b row c d end table close parentheses
    • Write (x, y) as a column vector, open parentheses table row x row y end table close parentheses
    • Use matrix multiplication to work out open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses, which gives open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses
    • Write down the image point coordinates, (x', y')
  • In harder questions you may be given the image coordinates, (x', y') and asked to find the original coordinates 
    • introduce letters (e.g. x and y) for the original coordinates, (x, y)
    • use the matrix M to set up and solve simultaneous equations to find x and y

Worked example

A matrix, M, is given by  bold M equals open parentheses table row 4 5 row 1 cell negative 2 end cell end table close parentheses.

(a)
Work out the coordinates of the image of the point open parentheses 2 comma space 3 close parentheses using the transformation represented by M.

 

Multiply the transformation matrix bold M by the coordinates, written as a column vector 

open parentheses table row 4 5 row 1 cell negative 2 end cell end table close parentheses open parentheses table row 2 row 3 end table close parentheses equals open parentheses table row cell 4 cross times 2 space plus space 5 cross times 3 end cell row cell 1 cross times 2 space plus space minus 2 cross times 3 end cell end table close parentheses equals open parentheses table row 23 row cell negative 4 end cell end table close parentheses  

Rewrite the answer as coordinates

begin bold style stretchy left parenthesis 23 comma negative 4 stretchy right parenthesis end style
  

(b)
The image of another point, P, using the transformation represented by M is open parentheses 11 comma space 6 close parentheses. 
Work out the coordinates of P.
 

You do not know the coordinates of P, so we can write it as open parentheses x comma y close parentheses  

Let P equals open parentheses x comma y close parentheses 

Multiply the transformation matrix bold M by the coordinates P, written as a column vector
This time, you know the image of the point after it is transformed, so can fill this in as the "answer"

table row cell open parentheses table row 4 5 row 1 cell negative 2 end cell end table close parentheses open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row 11 row 6 end table close parentheses end cell row cell open parentheses table row cell 4 x plus 5 y end cell row cell x minus 2 y end cell end table close parentheses end cell equals cell open parentheses table row 11 row 6 end table close parentheses end cell end table 

Equate the matching elements of the two matrices

table row cell 4 x plus 5 y end cell equals 11 row cell x minus 2 y end cell equals 6 end table 

You now have a pair of simultaneous equations which can be solved using either elimination or substitution

Use substitution to rearrange the second equation to make x the subject

x equals 2 y plus 6 

Substitute this into the first equation, and solve to find y 

table row cell 4 open parentheses 2 y plus 6 close parentheses plus 5 y end cell equals 11 row cell 8 y plus 24 plus 5 y end cell equals 11 row cell 13 y plus 24 end cell equals 11 row cell 13 y end cell equals cell negative 13 end cell row y equals cell negative 1 end cell end table 

Substitute y equals negative 1 into the second equation to find x 

table row cell x minus 2 open parentheses negative 1 close parentheses end cell equals 6 row cell x plus 2 end cell equals 6 row x equals 4 end table 

P is stretchy left parenthesis 4 comma negative 1 stretchy right parenthesis

Rotation Matrices

How do I find rotation 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 a rotation about the origin, A moves to A' and C moves to C' 
    • The matrix, 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' (as 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 I is the identity matrix

Worked example

The matrix M represents a rotation of 270° anticlockwise about the origin. 
Work out M.
 
A rotation of 270° anticlockwise is the same as a rotation of 90° clockwise

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

transforming-a-point-we

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

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

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 their new positions
        • 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.
 

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 

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

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 

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 about 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 their new positions
      • 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 about 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 about the origin is open parentheses table row cell negative 0.5 end cell 0 row 0 cell negative 0.5 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.
 
 
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 bold 1 over bold 4 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.