Representations

$D_8 = \langle r, s\mid r^4 = s^2 = e,\ srs = r^{-1}\rangle$ acts on $\mathbb{R}^2$ with $r$ the rotation by $\pi/2$ and $s$ the reflection in the $x$-axis. Write down the matrices $R, S$, check the defining relations, and compute the traces of $R$, $R^2$ and $S$.