General Fundamental Matrix

Theorem
Let $$\Phi(t)$$ be a fundamental matrix of the system $$x' = A(t)x$$.

Then
 * $$\Phi(t)C$$

is a general fundamental matrix of $$x' = A(t)x$$, where $$C$$ is any nonsingular matrix.

Proof
Clearly $$\Phi(t)C$$ is a fundamental matrix:


 * $$\frac{d}{dt} \Phi(t)C = \Phi'(t)C = A(t)\Phi(t)C$$;


 * $$\det( \Phi(t) C ) = \det(\Phi(t)) \det(C) \ne 0$$.

Let $$\Psi(t)$$ be an arbitrary fundamental matrix.

Then from General Vector Solution of Fundamental Matrix $$\Phi(t) \Phi^{-1}(t_0) \Psi(t_0)$$ solves the same matrix equation and has the same initial conditions.

Hence by uniqueness $$\Phi(t) \Phi^{-1}(t_0) \Psi(t_0)$$ is equal to $$\Psi(t)$$.

Letting $$C = \Phi^{-1}(t_0) \Psi(t_0)$$ finishes the proof.