  The block Kaczmarz method and its variants are designed for solving the
over-determined linear system. They involve iteratively projecting the current
point onto the solution space of a subset of constraints. In this work, by
alternately dealing with two subproblems (i.e., linear system with multiple
right-hand sides) using the block Kaczmarz method, we propose the {\it
Alternating Randomized Block Kaczmarz} (ARBK) method to solve the linear matrix
equation $AXB=F$, which incorporates a randomized index selection scheme to
determine the subset of constraints. The convergence analysis reveals that the
ARBK method has a linear convergence rate bounded by an explicit expression.
Several numerical studies have been conducted to validate the theoretical
findings.
