We study the critical behaviour of matrix models with builtin SU(2) geometry by using Hybrid Monte Carlo (HMC) techniques. The first system under study is a matrix regularization of the φ4 theory defined on the sphere. We develop a HMC algorithm together with an SU(2) gauge-fixing procedure in order to study the model. We extract the phase diagram of the model and give an estimation for the triple point for a system constructed of matrices of size N = 7. Our numerical results also suggest the existence of stripe phases- phases in which modes with higher momentum l have non-negligible contribution. The second system under study is a matrix model realized via competing Yang-Mills and Myers terms. In its low-temperature phase the system has geometrical phase with SO(3) symmetry: the ground state is represented by the su(2) generators. This geometry disappears in the high-temperature phase the system. Our results suggest that there are three main types of fluctuations in the system close to the transition: fluctuations of the fuzzy sphere, fluctuations which drive the system between the two phases, and fluctuations of the high-temperature regime. The fluctuations of the fuzzy sphere show the properties of a second order phase transition. We establish the validity of the finite size scaling ansatz in that regime. The fluctuations which bring the system between the phases show the properties of a first order transition. In the Appendix we provide in some detail the idea behind the HMC approach. We give some practical guidelines if one is to implement such an algorithm to study matrix models. We comment on the main sources for the phenomenon of autocorrelation time. As a final topic we present the basics of the OpenCL language which we used to port some of our algorithms for parallel computing architectures such as GPU’s.