We explicitly compute nonperturbative effects in a supersymmetric double-well matrix model corresponding to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. We analytically determine the full one-instanton contribution to the free energy and one-point function, including all perturbative fluctuations around the one-instanton background. The leading order two-instanton contribution is determined as well. We see that supersymmetry is spontaneously broken by instantons, and that the breaking persists after taking a double scaling limit which realizes the type IIA theory from the matrix model. The result implies that spontaneous supersymmetry breaking occurs by nonperturbative dynamics in the target space of the IIA theory. We also discuss recent progress in summing up all instanton contributions using knowledge of random matrix theory