Why is this group of matrices isomorphic to the dihedral group?

Question

I am reading Abstract Algebra, Theory and Applications by Judson and in exercise $13$ chapter $9$, Isomorphisms, I need to prove that the set of matrices $$A=\pmatrix{ \omega & 0 \\ 0 & \omega ^{-1}} \qquad B=\pmatrix{ 0 & 1 \\ 1 & 0}$$ Where $\omega = e^{2\pi i /n}$ form a group isomorphic to $D_n$ (dihedral group). I know how matrices work to reflect and rotate objects in a dihedral group, and I know that $$R=e^{2 \pi i /n} =\pmatrix{ \cos (\frac{2\pi i}{n}) & -\sin (\frac{2\pi i}{n}) \\ \sin (\frac{2\pi i}{n}) & \cos (\frac{2\pi i}{n})}$$ I can also see that $A^k$ will yield all the rotations I need, reaching the identity rotation with $A^n$, and I can see that multiplying by $B$ allows us to reach the reflections. It is clear that $A$ and $R$ are related, and even though it seems very logical, I cannot prove it. How are these two groups isomorphic?