I am approaching the theory of these kind of problems but I am missing an example.
I am tasked to solve:
$$X'= \left( \begin{array}{ccc} 1 & 4 \\ 1 & 1 \\ \end{array} \right)X + \left( \begin{array}{ccc} 1 \\ 1 \\ \end{array} \right)e^t , \ \ \ X(0) = \left( \begin{array}{ccc} 2 \\ 1 \\ \end{array} \right) $$
the general solution ,from my understanding, is:
$$X(t) = e^{At}X(0) + e^{At}\int_0^te^{-As}f(s) ds$$
I think $ e^{At } $ is equal to $X(t)X(0)^{-1}$ but how can I invert a column vector?
Moreover I am unsure what to substitute in $\int_0^te^{-As}f(s)ds$.
Could I have an example or a reference so I can then try my hand on this problem?