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Transformations using Matrices (Edexcel International AS Further Maths) : Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Transforming Points using Matrices

How do I transform a point using a matrix?

  • A point open parentheses x comma space y close parentheses in a 2D plane can be transformed (mapped) on to another point open parentheses x apostrophe comma space y apostrophe close parentheses by a 2 × 2 matrix, bold M

    • This is called a linear transformation

      • open parentheses x comma space y close parentheses is the object and open parentheses x apostrophe comma space y apostrophe close parentheses is the image

  • The coordinates of the image point can be found using matrix multiplication

  • To transform open parentheses x comma space y close parentheses by the matrix open parentheses table row a b row c d end table close parentheses

    • Write open parentheses x comma space y close parentheses 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, open parentheses x apostrophe comma space y apostrophe close parentheses

  • If given image coordinates, open parentheses x apostrophe comma space y apostrophe close parentheses, and asked to find original coordinates 

    • use inverse matrices

    • bold M open parentheses table row x row y end table close parentheses equals open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses space space space rightwards double arrow space 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

How do I transform a set of vertices?

  • Vertices of a shape can be transformed one-by-one using the method above

    • This gives the vertices of the new shape

      • Straight lines between the vertices will still be straight lines in the new shape

    • Sometimes the order (clockwise or anticlockwise) of the vertices is reversed

      • This is also called changing the sense of the vertices

  • An alternative method is to stack column vectors of coordinates into a vertex matrix

    • open parentheses 1 comma space 2 close parentheses, open parentheses 3 comma space 4 close parentheses, open parentheses 5 comma space 6 close parentheses and open parentheses 0 comma space 10 close parentheses become open parentheses table row 1 3 5 0 row 2 4 6 10 end table close parentheses

    • Then multiply the matrix bold M by the vertex matrix above

    • The new columns represent the corresponding image coordinates

Worked Example

The 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 triangle has coordinates open parentheses 1 comma space 0 close parentheses, open parentheses 2 comma space 3 close parentheses and open parentheses negative 1 comma space 5 close parentheses.

Work out the coordinates of the triangle after the transformation represented by the matrix bold M has been applied.

 

Multiply the matrix bold M by each set of coordinates, written as a column vectors

table row cell open parentheses table row 4 5 row 1 cell negative 2 end cell end table close parentheses open parentheses table row 1 row 0 end table close parentheses end cell equals cell open parentheses table row cell 4 cross times 1 space plus space 5 cross times 0 end cell row cell 1 cross times 1 space plus space minus 2 cross times 0 end cell end table close parentheses equals open parentheses table row 4 row 1 end table close parentheses end cell row cell 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 end cell equals cell 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 end cell row cell open parentheses table row 4 5 row 1 cell negative 2 end cell end table close parentheses open parentheses table row cell negative 1 end cell row 5 end table close parentheses end cell equals cell open parentheses table row cell 4 cross times open parentheses negative 1 close parentheses space plus space 5 cross times 5 end cell row cell 1 cross times open parentheses negative 1 close parentheses space plus space minus 2 cross times 5 end cell end table close parentheses equals open parentheses table row 21 row cell negative 11 end cell end table close parentheses end cell end table   

Rewrite the answers as coordinates

open parentheses 4 comma space 1 close parentheses, open parentheses 23 comma space minus 4 close parentheses and open parentheses 21 comma space minus 11 close parentheses

  Alternatively, you can multiply bold M by a vertex matrix: open parentheses table row 4 5 row 1 cell negative 2 end cell end table close parentheses open parentheses table row 1 2 cell negative 1 end cell row 0 3 5 end table close parentheses equals open parentheses table row 4 23 21 row 1 cell negative 4 end cell cell negative 11 end cell end table close parentheses

Determinants as Area Scale Factors

  • When transforming vertices of an object using the matrix bold M to give vertices of the image

    • the magnitude of the determinant of bold M is the area scale factor of enlargement from the object to the image

    • vertical line det open parentheses bold M close parentheses vertical lineis the area scale factor

  • For example, if a shape is transformed by bold M equals open parentheses table row 1 3 row 2 4 end table close parentheses then

    • det open parentheses bold M close parentheses equals 1 cross times 4 minus 2 cross times 3 equals negative 2

    • sovertical line det open parentheses bold M close parentheses vertical line equals vertical line minus 2 vertical line equals 2

      • This transformation doubles the area of any shape

  • A negative determinant means the sense (clockwise or anticlockwise) of the vertices is reversed

Examiner Tips and Tricks

Remember the modulus signs around the determinant as this can often lead to two solutions in algebraic questions.

Worked Example

A transformation, represented by the matrix bold M equals open parentheses table row 3 2 row 4 cell space space space k plus 1 end cell end table close parentheses, where k is a constant, maps a quadrilateral of area 45 square units to a quadrilateral of area 450 square units.

Find the possible values of k.

Find the scale factor of enlargement of the area from 45 to 450

Area scale factor is 10

The magnitude of the determinant is the area scale factor
Start by finding and simplifying the determinant of bold M

table row cell det open parentheses bold M close parentheses end cell equals cell 3 open parentheses k plus 1 close parentheses minus 2 cross times 4 end cell row blank equals cell 3 k plus 3 minus 8 end cell row blank equals cell 3 k minus 5 end cell end table

Set the magnitude of the determinant equal to 10

vertical line 3 k minus 5 vertical line equals 10

This means 3 k minus 5 is either equal to 10 or -10
Solve each equation

table row cell 3 k minus 5 end cell equals 10 row cell 3 k end cell equals 15 row k equals 5 end table

or

table row cell 3 k minus 5 end cell equals cell negative 10 end cell row cell 3 k end cell equals cell negative 5 end cell row k equals cell negative 5 over 3 end cell end table

Write out the two answers

k equals 5 space space or space space k equals negative 5 over 3

If the question had said "where k is an integer" then only k equals 5 would be accepted

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.