  We study a matrix model that has $\phi_a^i\ (a=1,2,\ldots,N,\
i=1,2,\ldots,R)$ as its dynamical variable, whose lower indices are pairwise
contracted, but upper ones are not always done so. This matrix model has a
motivation from a tensor model for quantum gravity, and is also related to the
physics of glasses, because it has the same form as what appears in the replica
trick of the spherical $p$-spin model for spin glasses, though the parameter
range of our interest is different. To study the dynamics, which in general
depends on $N$ and $R$, we perform Monte Carlo simulations and compare with
some analytical computations in the leading and the next-leading orders. A
transition region has been found around $R\sim N^2/2$, which matches a relation
required by the consistency of the tensor model. The simulation and the
analytical computations agree well outside the transition region, but not in
this region, implying that some relevant configurations are not properly
included by the analytical computations. With a motivation coming from the
tensor model, we also study the persistent homology of the configurations
generated in the simulations, and have observed its gradual change from $S^1$
to higher dimensional cycles with the increase of $R$ around the transition
region.
