Eigenvalues · Hard problem

A plane figure is redrawn every second by the linear map $A = \begin{pmatrix}1 & -1\\1 & 1\end{pmatrix}$, which rotates and scales it (its eigenvalues are non-real). After how many whole seconds does the figure FIRST point in exactly its original direction again (i.e. $A^n$ is a positive scalar multiple of the identity)?