Question

Let
\(A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix}\)
and
\(B = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.\)

Let \(S\) be the set of all matrices that can be generated from taking products of \(A\) and \(B\), in any order. For example, the matrix \(A^2B A^4B\) is in the set \(S\). Find the number of distinct elements in \(S\).