  In this thesis, we studied a Gaussian approximation to the bosonic part of
the BFSS matrix model using Monte Carlo simulations based on the Metropolis
algorithm. We reproduce with great accuracy the stringy Hagedorn phase
transition from a confinement (black string) phase to a deconfinement (black
hole) phase. We used the Polyakov loop as an order parameter to investigate the
large-N behavior of this model at different temperatures, other observables
such as internal energy and extent of space were also computed. In the last
part, we present the matrix-geometry approach to a modified action where we
captured only a remnant of the geometric Yang-Mills to a baby-fuzzy-sphere
phase where the fuzzy sphere solution is only manifested as a three-cut
configuration. The Yang-Mills phase retains most of its characteristics with
two exceptions: i) the uniform distribution inside a solid ball suffers a
crossover at very small values of the gauge coupling constant to a Wigner's
semi-circle law, and ii) the uniform distribution at small T is non-existent.
