Next use Gauss-Jordan elimination to transform the left-hand 3X3 side of your augmented matrix into the identity matrix, thus:
The resulting 3X3 on the right, consisting of through will be your identity matrix.
Fortunately, there is a much easier way. Go to http://www.bluebit.gr/matrix-calculator/, enter your matrix just like you did here separating the elements with spaces and the rows with carriage returns, check off Matrix Inverse, and then click Calculate.