Diagonalization & Matrix Powers · Challenge problem

A two-mass system's displacement vector $\mathbf x(t)$ satisfies $\mathbf x'' = A\mathbf x$ with $A = \begin{pmatrix}-13/2 & 5/2\\5/2 & -13/2\end{pmatrix}$, released from rest ($\mathbf x'(0)=\mathbf 0$) at $\mathbf x(0) = \begin{pmatrix}3\\1\end{pmatrix}$. Diagonalizing $A$ splits the motion into two independent oscillation modes with different periods. Find the first time $t>0$ at which the system returns EXACTLY to its initial position $\begin{pmatrix}3\\1\end{pmatrix}$.