Challenge Level

This problem involves the algebra of matrices and various geometric concepts associated with vectors and matrices. As you consider each point, make use of geometric or algebraic arguments as appropriate. If there is no definitive answer to a given part, try to give examples of when the question posed is or is not true.

In the five questions below: $R, S$ are rotation matrices; $P, Q$ are reflection matrices; $M,N$ are neither rotations nor reflections. Consider each part in 2D and 3D.

- Is it always the case that $M+N = N + M$?
- It it always the case that $RS= SR$?
- It it always the case that $RP= PR$?
- It it always the case that $PQ= QP$?
- Is it ever the case that $MN = NM$?

How do the values of the determinants of the various matrices
affect the results of these questions?