A plane reflection has determinant $-1$, while a <rotation matrix> has determinant $+1$. The determinant of their product, in either order, is therefore $-1$. Since every rotation has determinant $+1$, such a product can never be another rotation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 8B / b / iii / Solution

 Solution (source code)