Stability of trivial solution for DE system with non-constant coefficient matrix 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.

Here is an example. 
$$
A(t)=\left(
\begin{matrix}
-1+\frac32\cos^2 t& 1-\frac32\cos t\sin t\\
-1-\frac32\sin t\cos t &-1+\frac32\sin^2 t
\end{matrix}
\right)
$$
One can check that the eigenvalues of $A(t)$ are
$$
\lambda_1=\frac14[-1+\sqrt7 i],\quad \lambda_2=\bar\lambda_1,
$$
both of which have negative real parts. But the origin is unstable. 
This is an example from the section about Floquet theory in Ordinary Differential Equations and Dynamical Systems by Sideris. 
