  The compressibility of magnetized quark matter is investigated in the SU(2)
NJL model. The increases of the chemical potential and the temperature can
reduce the compressibility, and lead to the much stiffer equation of state. The
variation of the compressibility with the magnetic field will depend on the
phase region. Due to the anisotropic structure, the compressibility is
different in the directions parallel and perpendicular to the field. The
discontinuity of longitudinal compressibility with the chemical potential and
the temperature captures the signature of a first-order chiral phase transition
and the crossover at high temperature. Moreover, the
magnetic-field-and-temperature running coupling would have an important effect
on the position of the phase transition. Under the lowest landau level
approximation at zero temperature, the longitudinal compressibility has a
direct inverse proportional relation to the magnetic field strength and the
chemical potential square as $\kappa^\parallel_{\mathrm{LLL},\chi}\propto1/(eB
\mu^2)$.
