The IKKT matrix model is a promising candidate of nonperturbative formulation of superstring theory. Since the Lorentzian version is not well-defined as it is, as opposed to the Euclidean version, it has been proposed recently to introduce a Lorentz invariant mass term as a regularization. Depending on the sign of this term, we obtain either a model equivalent to the Euclidean model due to an appropriate contour deformation, or a model which is genuinely Lorentzian. After reviewing these issues, we argue that only the genuinely Lorentzian model allows nontrivial classical solutions, which actually make the partition function diverge due to the noncompactness of the Lorentz symmetry that these solutions are not endowed with. This is confirmed explicitly in the simplest possible case of the N=2 bosonic model by Monte Carlo simulation and the 1/D expansion. We also discuss the impact of this statement in the possible emergence of (3+1)D expanding space-time at large N in the presence of supersymmetry.