  We study a family of monomial ideals, called block diagonal matching field
ideals, which arise as monomial Gr\"obner degenerations of determinantal
ideals. Our focus is on the minimal free resolutions of these ideals and all of
their powers. Initially, we establish their linear quotient property and
compute their Betti numbers, illustrating that their minimal free resolution is
supported on a regular CW complex. Our proof relies on the results of Herzog
and Takayama, demonstrating that ideals with a linear quotient property have a
minimal free resolution, and on the construction by Dochtermann and Mohammadi
of cellular realizations of these resolutions. We begin by proving the linear
quotient property for each power of such an ideal. Subsequently, we show that
their corresponding decomposition map is regular, resulting in a minimal
cellular resolution. Finally, we demonstrate that distinct decomposition maps
lead to different cellular complexes with the same face numbers.
