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Characteristics of the gyrotropic-nihility state are studied in a magnetic superlattice, which is influenced by an external static magnetic field. Investigations are carried out with the assistance of the effective medium theory, according to which the studied structure is approximated as a uniform gyroelectromagnetic medium. The theory of the gyrotropic-nihility state is developed in terms of the eigenwaves propagation in such gyroelectromagnetic medium. It turns out that in the frequency band around the frequency of gyrotropic-nihility state the studied structure appears to be matched to free space with both the refractive index and the wave impedance.