Inverse Matrix Transformations (Edexcel International AS Further Maths)

Revision Note

Written by: Mark Curtis

Reviewed by: Dan Finlay

If the matrix $M$ transforms the point P to P’ then
- the inverse matrix $M^{- 1}$ 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
- ${MM}^{- 1} = M^{- 1} M = I$

You can often interpret inverse matrix transformations geometrically
For example, let $M$ represent a rotation of 90° clockwise
- Then $M^{- 1}$ must represent a rotation of 90° anticlockwise
- $M^{- 1}$ is also the same as a rotation of 270° clockwise ( $M^{3}$ )
  - So $M^{3} = M^{- 1}$ giving $M^{4} = I$
  - This says four rotations of 90° clockwise returns to the original position
If $M = M^{- 1}$ 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!
- $M = M^{- 1}$ also gives $M^{2} = I$
  - This says reflecting twice about the y-axis returns to the original position

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

(a) Describe fully the single transformation represented by $Q^{- 1}$ .

The inverse of $Q$ reverses the transformation

$Q^{- 1}$ represents a rotation of 120° clockwise about the origin

(b) Use a geometrical argument to explain why $Q^{- 1} = Q^{2}$ .

$Q^{2}$ means apply the transformation represented by $Q$ twice

$Q^{2}$ 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, $Q^{- 1}$ , is the same as doing a rotation of 240° anticlockwise about the origin, $Q^{2}$

Last updated: 10 April 2024

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