Definition of terms:
Since we want to consider the possibility that there is also a translation from the centre, and that the rotation may not be about the centre but may be about some arbitrary point, then the matrix has to be increased from 3x3 to 4x4 so that it can specify the position and orientation. (as explained here).
The Multivector is based on 3D vectors and it can specify the position and orientation.
I am still working on this - can you help?