  Using variational methods, we numerically investigate the matrix model for
the two-color QCD coupled to a single quark (matrix-QCD$_{2,1}$) in the limit
of ultra-strong Yang-Mills coupling ($g =\infty$). The spectrum of the model
has superselection sectors labelled by baryon number $B$ and spin $J$. We study
sectors with $B=0,1,2$ and $J=0,1$, which may be organized as mesons,
(anti-)diquarks and (anti-)tetraquarks. For each of these sectors, we study the
properties of the respective ground states in both chiral and heavy quark
limits, and uncover a rich quantum phase transition (QPT) structure. We also
investigate the division of the total spin between the glue and the quark and
show that glue contribution is significant for several of these sectors. For
the $(B,J)=(0,0)$ sector, we find that the dominant glue contribution to the
ground state comes from reducible connections. Finally, in the presence of
non-trivial baryon chemical potential $\mu$, we construct the phase diagram of
the model. For sufficiently large $\mu$, we find that the ground state of the
theory may have non-zero spin.
