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Inverse Matrix Transformations (Edexcel International AS Further Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Inverse Matrix Transformations

What are inverse matrix transformations?

  • If the matrix bold M transforms the point P to P’ then

    • the inverse matrix bold M to the power of negative 1 end exponent transforms P’ back to P

  • Inverse matrices represent the reverse of the transformation

  • Applying a transformation then its inverse returns points to their original positions

    • bold MM to the power of negative 1 end exponent equals bold M to the power of negative 1 end exponent bold M equals bold I

How do I use inverse matrix transformations?

  • You can often interpret inverse matrix transformations geometrically

  • For example, let bold M represent a rotation of 90° clockwise

    • Then bold M to the power of negative 1 end exponent must represent a rotation of 90° anticlockwise

    • bold M to the power of negative 1 end exponent is also the same as a rotation of 270° clockwise (bold M cubed)

      • So bold M cubed equals bold M to the power of negative 1 end exponent giving bold M to the power of 4 equals bold I

      • This says four rotations of 90° clockwise returns to the original position

  • If bold M equals bold M to the power of negative 1 end exponent then the inverse does the same thing as the transformation

    • For example. the reverse of reflecting in the y-axis is reflecting in the y-axis!

    • bold M equals bold M to the power of negative 1 end exponent also gives bold M squared equals bold I

      • This says reflecting twice about the y-axis returns to the original position 

Worked Example

The matrix bold Q represents a rotation of 120° anticlockwise about the origin.

(a) Describe fully the single transformation represented by bold Q to the power of negative 1 end exponent.

The inverse of bold Q reverses the transformation

bold Q to the power of negative 1 end exponentrepresents a rotation of 120° clockwise about the origin

(b) Use a geometrical argument to explain why bold Q to the power of negative 1 end exponent equals bold Q squared.

bold Q to the power of bold 2 means apply the transformation represented by bold Q twice

bold Q to the power of bold 2 represents a rotation of 240° anticlockwise about the origin

Rotating 240° anticlockwise is the same as rotating 120° clockwise

A rotation of 120° clockwise about the origin, bold Q to the power of negative 1 end exponent, is the same as doing a rotation of 240° anticlockwise about the origin, bold Q squared

 

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    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

    Author: 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.