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We consider a $2\times 2$ system of one-dimensional semiclassical Schrödinger operators with small interactions with respect to the semiclassical parameter $h$. We study the asymptotics in the semiclassical limit of the resonances near a non-trapping energy for both corresponding classical Hamiltonians. We show the existence of resonances of width $T^{-1}h\log (1/h)$, contrary to the scalar case, under the condition that two classical trajectories cross and compose a periodic trajectory with period $T$.