We present a result of absence of absolutely continuous spectrum in an interval of $\R$, for a matrix-valued random Schrödinger operator, acting on $L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction potential is generic in the real symmetric matrices. For this purpose, we prove the existence of an interval of energies on which we have separability and positivity of the $N$ non-negative Lyapunov exponents of the operator. The method, based upon the formalism of Fürstenberg and a result of Lie group theory due to Breuillard and Gelander, allows an explicit contruction of the wanted interval of energies.