Matrices: Rotations as products of reflections in $\Bbb R^2$ Ok so I have two matrices:
The reflection about the line y=x. 
$$A=\pmatrix{0 & 1 \\ 1 & 0}$$
And the reflection about the line $y=0$.
$$B=
\pmatrix{
1 & 0 \\
0 & -1}
$$
I need to show that both $AB$ and $BA$ represent rotations of $\mathbb{R}^2$. I know that they are rotations of 90 degrees in opposite directions, but how can I show this? I don't think giving a few examples is enough.
Also, if I need to compute $ABABABAB$ and $BABABABA$, can I just bunch them up as $(AB)(AB)AB)(AB)$ and $(BA)(BA)(BA)(BA)$?

$$AB = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \ \ \textbf{and}\ \ BA=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$
Now take any vector $\textbf{x} = (u,v)$ then $AB(u,v) \cdot (u,v) = 0$ and $BA(u,v) \cdot (u,v) = 0$. Hence what can you say about the matrices since the vectors they give are always perpendicular? Once you see what their rotations correspond to, you can easily compute the matrix order. 
