# Definition:Fundamental Matrix

## Definition

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 $\mathbf x' = A \left({t}\right) \mathbf x$ if and only if:

it solves the matrix system $\mathbf X'=A(t) \mathbf X$
$\det \Phi \left({t}\right)$ is nonvanishing