Suppose that
$$ A = 
 \begin{bmatrix}
 0 & 2 \\
 2 & 0 \\
 \end{bmatrix}
$$

Then $e^{At} = I \cos(2t)+ \frac{A}{2} \sin(2t)$.  

How is this fact applied to a **general solution of $x' = Ax$** using the Eigenvalue method?