  We explore a family of monomial ideals derived as Gr\"obner degenerations of
determinantal ideals. These ideals, previously examined as block diagonal
matching field ideals within the realm of toric degenerations of Grassmannians,
are identified as monomial initial ideals of determinantal ideals. We establish
their linear quotient property and calculate their Betti numbers, demonstrating
that their minimal free resolution is supported on a regular CW complex,
equivalently, they have a cellular resolution. On the combinatorial front, we
examine a range of weight vectors that yield diverse monomial initial ideals
for determinantal ideals. We show that despite variations in weight vectors,
all these ideals share identical Betti numbers. However, altering a weight
vector results in distinct cellular complexes supporting their minimal free
resolutions.
