Determination of the state-transition matrix

Question

I am trying to find the state-transition matrix (STM) via different techniques $X'= AX$ where $A$ is time-invariant.
where $\phi$ is the STM. $$\phi '=A \phi$$ Once we take the derivative of $\phi '=A \phi$ we obtain $$\phi'' = \text{A}*\phi'..... (1)$$ or $$ \phi''=A*A \phi.... (2) $$

Note $\text A$ is a constant matrix.

So the question is can I numerically integrate equation (1) or (2) with initial conditions on $\phi(0)$ and $\phi'(0)$ and obtain STM?

I tried it using Mathematica but answers do not match. Thanks Code and output

Answer 1

Given ${\bf A} \in \Bbb R^{n \times n}$, the state-transition matrix satisfies the following matrix differential equation

$$ \dot {\bf X} (t) = {\bf A} {\bf X} (t) $$

whose solution is ${\bf X} (t) = \exp( t {\bf A} ) \, {\bf X} (0)$.