Further Concepts for Advanced Mathematics - FP1
Unit 2Matrices – Section2gThe Inverse of a 2 x 2 Matrix
Transformations and their Inverses
1) Shear
The matrixrepresents a shear parallel to the x axis.
The equations of the transformation are
The inverse of the matrix would transform the image back to the object.
It is obvious that the equations of this transformation would be .
The inverse would therefore be .
2) Rotation
The matrix represents a rotation of anticlockwise about the origin.
The equations for this transformation are
The equations to transform back to the object are therefore
The inverse matrix of is .
From these two examples you can see that two of the values in the inverse matrix are the opposite sign to those in the originals matrix. We can use this idea to speed up finding inverse matrices.
3) A different transformation
The transformation represented by the matrix is shown on the diagram below (the object is the unit square):
If we call the inverse of the transformation matrixthen we can use the fact that applying it to the image would get us back to the object so
Leading to the equations , , , , and
From these, we can find that , , and
This means that the inverse of is . This could be better written by taking out a factor of to give the inverse as .
The value of the fraction is significant here. It is the same as the determinant of the original matrix. You can also see that two of the diagonal values have swapped places and the other two have changed their signs.
The general form
We can use the ideas above to write the inverse of any 2 x 2 matrix.
For the matrix M where , the inverse is M-1 where
Some properties
You can test these with
The Inverse of a Product
This can be interpreted geometrically by considering AB to be the transformation B followed by the transformation A. To go backwards through this we would first have to use the inverse of A and then the inverse of B hence B-1A-1.
1