Definition:Fundamental Matrix

A matrix function $$\Phi(t)$$ is a fundamental matrix of the system $$x'=A(t)x$$ if it solves the matrix system $$X'=A(t)X$$ and $$\det \Phi(t)$$ is nonvanishing.

See:
 * General Vector Solution of Fundamental Matrix
 * General Fundamental Matrix

for results concerning the general solution to such a system of equations.