  (i) For the matrix Schr\"{o}dinger operator on the half line, it is shown
that if the potential exponentially decreases fast enough then only the
scattering matrix uniquely determines the self-adjoint potential and the
boundary condition. (ii) For the matrix Schr\"{o}dinger operator on the full
line, it is shown that if the potential exponentially decreases fast enough
then the scattering matrix (or equivalently, the transmission coefficient and
reflection coefficient) uniquely determine the potential. If the potential
vanishes on $(-\infty,0)$ then only the left reflection coefficient uniquely
determine the potential.
