IGCSEMathsEdexcelMaths BRevision NotesMatricesTransformations using MatricesTransforming a Point

Transforming a Point (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Transforming a point

How do I transform a point using a matrix?

  • A point (x, y) in a 2D plane can be transformed onto another point (x',y') by a matrix, bold 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

      • Note that this is the same as a 2 cross times 1 matrix

    • 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

      • open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row cell a x plus b y end cell row cell c x plus d y 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)

      • then use the matrix bold M to set up and solve simultaneous equations to find x and y

    • Alternatively, you could use the inverse matrix bold M to the power of negative 1 end exponent to undo the transformation

      • table row cell bold M open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses space space left right double arrow space space open parentheses table row x row y end table close parentheses equals bold M to the power of negative 1 end exponent open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell end table

      • You can see this by solving the matrix equation for open parentheses table row x row y end table close parentheses:

table row cell bold M open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell row cell bold M to the power of negative 1 end exponent bold M open parentheses table row x row y end table close parentheses end cell equals cell bold M to the power of negative 1 end exponent open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell row cell bold I open parentheses table row x row y end table close parentheses end cell equals cell bold M to the power of negative 1 end exponent open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell row cell open parentheses table row x row y end table close parentheses end cell equals cell bold M to the power of negative 1 end exponent open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell end table

Worked Example

A matrix, bold 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 bold M.

Answer:

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

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

Error converting from MathML to accessible text.
  

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

Work out the coordinates of P.

Answer:

Method 1

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

Let P equals open parentheses x comma y close parentheses 

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

  • 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

Using substitution, start by rearranging 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 Error converting from MathML to accessible text.

Method 2

Find the inverse of matrix bold M

  • The inverse of space open parentheses table row a b row c d end table close parentheses space is space fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses

table row cell bold M to the power of bold minus bold 1 end exponent end cell equals cell fraction numerator 1 over denominator 4 cross times open parentheses negative 2 close parentheses minus 5 cross times 1 end fraction open parentheses table row cell negative 2 end cell cell negative 5 end cell row cell negative 1 end cell 4 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 13 end fraction open parentheses table row cell negative 2 end cell cell negative 5 end cell row cell negative 1 end cell 4 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 over 13 end cell cell 5 over 13 end cell row cell 1 over 13 end cell cell negative 4 over 13 end cell end table close parentheses end cell end table

If bold space bold M open parentheses table row x row y end table close parentheses equals open parentheses table row 11 row 6 end table close parentheses, then space open parentheses table row x row y end table close parentheses equals bold M to the power of negative 1 end exponent open parentheses table row 11 row 6 end table close parentheses

table row cell open parentheses table row x row y end table close parentheses end cell equals cell open parentheses table row cell 2 over 13 end cell cell 5 over 13 end cell row cell 1 over 13 end cell cell negative 4 over 13 end cell end table close parentheses open parentheses table row 11 row 6 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 over 13 cross times 11 plus 5 over 13 cross times 6 end cell row cell 1 over 13 cross times 11 plus open parentheses negative 4 over 13 close parentheses cross times 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 52 over 13 end cell row cell negative 13 over 13 end cell end table close parentheses end cell row blank equals cell open parentheses table row 4 row cell negative 1 end cell end table close parentheses end cell end table

Write as coordinates

P is Error converting from MathML to accessible text.

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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.