The semiclassical limit $\varepsilon \to 0$ of the scattering matrix S associated with the equation $i\varepsilon \frac{{d\varphi (t)}} {{dt}} = A(t)\varphi (t)$ is considered. If $A(x)$ is an analytic $n \times n$ matrix whose eigenvalues are real andnondegenerate for all $x \in {\bf R}$, the matrix S is computed asymptotically up to errors $O(e^{\kappa \varepsilon ^{ - 1} } )$, $\kappa > 0$. Moreover, for the case $n = 2$ and under further assumptions on the behavior of the analytic continuations of the eigenvalues of $A(x)$, the exponentially small off diagonal elements of S are given by an asymptotic expression accurate up to relative errors $O(e^{\kappa \varepsilon ^{ - 1} } )$. The adiabatic transition probability for the time-dependent Schrödinger equation, the semiclassical above barrier reflection coefficient for the stationary Schrödinger equation, and the total variation of the adiabatic invariant of a time-dependent classical oscillator are computed asymptotically to illustrate results.