\begin{definition}[Definition:Fundamental Matrix]
Let $\mathbf x' = A \left({t}\right) \mathbf x$ be a system of $n$ linear first order ODEs.
Let $\Phi \left({t}\right)$ be an $n \times n$ matrix function.
Then $\Phi \left({t}\right)$ is a '''fundamental matrix''' of the system $\mathbf x' = A \left({t}\right) \mathbf x$ {{iff}}:
:it solves the matrix system $\mathbf X'=A(t) \mathbf X$
:$\det \Phi \left({t}\right)$ is nonvanishing
\end{definition}