Inverse Matrix Transformations (Edexcel International AS Further Maths: Further Pure 1): Revision Note

Inverse Matrix Transformations

What are inverse matrix transformations?

• If the matrix $\mathbf{M}$ transforms the point P to P’ then

  • the inverse matrix ${\mathbf{M}}^{-1}$ 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

  • ${\mathbf{MM}}^{-1}={\mathbf{M}}^{-1}\mathbf{M}=\mathbf{I}$

How do I use inverse matrix transformations?

• You can often interpret inverse matrix transformations geometrically

• For example, let $\mathbf{M}$ represent a rotation of 90° clockwise

  • Then ${\mathbf{M}}^{-1}$ must represent a rotation of 90° anticlockwise

  • ${\mathbf{M}}^{-1}$ is also the same as a rotation of 270° clockwise (${\mathbf{M}}^3$)

    • So ${\mathbf{M}}^3={\mathbf{M}}^{-1}$ giving ${\mathbf{M}}^4=\mathbf{I}$

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

• If $\mathbf{M}={\mathbf{M}}^{-1}$ 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!

  • $\mathbf{M}={\mathbf{M}}^{-1}$ also gives ${\mathbf{M}}^2=\mathbf{I}$

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