In this subsection we show how the inverse of a matrix can be obtained (if it exists).
Form the matrix products and where
You will see that had we chosen instead of then both products and will be equal to . This requires . Hence this matrix is the inverse of . However, note, that if then has no inverse . (Note that for the matrix which occurred in the last task, confirming, as we found, that has no inverse.)
The Inverse of a 2 2 Matrix
If then the matrix has a (unique) inverse given by
Note that , the determinant of the matrix .
In words: To find the inverse of a matrix we interchange the diagonal elements, change the sign of the other two elements, and then divide by the determinant of .
Which of the following matrices has an inverse?
Therefore, and each has an inverse. does not because it has a zero determinant.
Find the inverses of the matrices and in the previous Task.
Use Key Point 8:
It can be shown that the matrix represents an anti-clockwise rotation through an angle in an -plane about the origin. The matrix represents a rotation clockwise through an angle . It is given therefore by
Form the products and for these ‘rotation matrices’. Confirm that is the inverse matrix of .
Effectively: a rotation through an angle followed by a rotation through angle is equivalent to zero rotation.