$\newcommand{\ones}{\mathbf 1}$

For $A \in {\mathbf R}^{2\times 2}$, \[ e^{tA} = \left[ \begin{array}{cc} e^{a_{11}t} & e^{a_{12}t} \\ e^{a_{21}t} & e^{a_{22}t} \end{array} \right]. \]


Suppose $\dot x(t) =Ax(t)$.

$x(5.2) = \exp (1.2 A ) x(4.0)$.

$x(5.2) = \exp (-1.2 A ) x(6.4)$.


Suppose $A \in {\mathbf R}^{n\times n}$, with resolvent $R= (sI-A)^{-1}$.

Every pole of $R$ is an eigenvalue of $A$.

Every eigenvalue of $A$ is a pole of $R$.