Given the arbitrary linear system of DE's $$x'=A(t)x,$$ with the condition that the spectral bound of $A(t) $ is uniformly bounded by a negative constant, is the trivial solution always stable? All the $(2\times 2)$ matrices I've tried which satisfy the above property yield stable trivial solutions, which seems to suggest this might be the case in general. I can't think of a simple counterexample, so I'm asking if one exists. If there isn't what would be some steps toward proving the statement?
This is indeed homework.