Consider the system of $n$ linear equations $x'=Ax$. Suppose the system conserved the quantity $$\sum_{k=1}^n a_k x_k(t)=C \ \ \ \ \ (\star) $$ for some $C \in \mathbb{R}.$ for some $a =(a_1,...,a_k)\in \mathbb{R}^n$. Prove that $A$ has a zero eigenvalue.
I do not know even how to start with this less information, I feel that the vector $a$ causing the existence of the zero eigenvalue, but how to show that. I appreciate any help or hints with that.
Thank you.
Hint: Differentiating both sides of $(\star)$, you get
$$ \sum_{k=1}^{n} a_{k} x'_{k}(t)= \sum_{k=1}^{n} a_k [Ax(t)]_k = \mathbf a \cdot A\mathbf x(t)=0. $$