Various classical solutions to lower dimensional Ishibashi-Kawai-Kitazawa-Tsuchiya-like Lorentzian matrix models are examined in their commutative limit. Poisson manifolds emerge in this limit, and their associated induced and effective metrics are computed. Signature change is found to be a common feature of these manifolds when quadratic and cubic terms are included in the bosonic action. In fact, a single manifold may exhibit multiple signature changes. Regions with a Lorentzian signature may serve as toy models for cosmological spacetimes, complete with cosmological singularities, occurring at the signature change. The singularities are resolved away from the commutative limit. Toy models of open and closed cosmological spacetimes are given in two and four dimensions. The four-dimensional cosmologies are constructed from noncommutative complex projective spaces, and they are found to display a rapid expansion near the initial singularity.