Inverse Matrix Transformations (Edexcel IAL Further Maths)

Spec point — 6.4

Inverse Matrix Transformations

What are inverse matrix transformations?

• 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

  • $\mathrm{MM}^{-1}=M^{-1}M=I$

How do I use inverse matrix transformations?

• 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

Worked Example 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

Final answer: $Q^{-1}$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 equation$Q$ twice

$Q^{2}$ represents a rotation of 240° anticlockwise about the origin

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

Final answer: A rotation of 120° clockwise about the origin,$Q^{-1}$, is the same as doing a rotation of 240° anticlockwise about the origin,$Q^{2}$