Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 6B (Vectors and Matrices)

An anticlockwise rotation matrix and reflection in the $x$-axis areThe columns of $R$ are the images of the standard coordinate vectors, while $M(x,y{)}^T=(x,-y{)}^T$.

Direct multiplication givesHenceThis is the defining conjugation relation between a rotation and reflection in a dihedral group.

No. Every power $R^a$ has determinant $1$, whereasMatrices with different determinants cannot be equal.

The relationsimply $RM=M{R}^{-1}$. Moving every occurrence of $M$ to the left and reducing exponents therefore puts every word into one of the normal formsExplicitly, with $\theta =2\pi /n$,andThe $n$ rotations are distinct. The $n$ reflected matrices are also distinct, and no reflected matrix equals a rotation because their determinants are $-1$ and $1$, respectively. Hence there are exactlymatrices, forming the dihedral group.