You probably didn't try it everywhere then. To be clear, when you did this,
R1R2 = R1 . R2 // MatrixForm
You assigned to R1R1 an expression with head MatrixForm. Try evaluating this:
FullForm[R1R2]
There is no built in arithmetic defined for MatrixForm. MatrixForm is just a wrapper for display purposes. The convention I use when I want to see a nice representation of a thing is to separate the definition from the display:
R1R2 = R1 . R2; R1R2 // MatrixForm
If you do this throughout your notebook, you'll get what you're looking for.
