Let $A \in M(n)$, let $\lambda \in \mathbb{R}$, let $V_{\lambda} := \ker(\lambda I - A)$ and let $x:\mathbb{R} \to \mathbb{R}^{n}$ be a solution of $\dot x= Ax$ such that $x(t_0) \in V_{\lambda}$ for some $t_0 \in \mathbb{R}$, then $x(t) \in V_{\lambda}$ for every real $t \in \mathbb{R}$.
Since $x(t_0) \in V_{\lambda}$, then $\dot x(t_0)= \lambda x(t_0)$, given that $\dot x(t_0) = Ax(t_0)$. But I don't know how to go on!