  Layered perovskites $A_3M_2$O$_7$ are known to exhibit the so-called hybrid
improper ferroelectricity. Despite experimentally confirmed cases (e.g.
nonmagnetic $M$=Ti and Sn), the ferroelectricity in magnetic Ca$_3$Mn$_2$O$_7$
remains a puzzle. Here, the structural, ferroelectric, magnetoelectric, and
optical properties of Ca$_3$Mn$_2$O$_7$ are systematically investigated.
Switchable polarization is directly measured, demonstrating its
ferroelectricity. In addition, magnetoelectric response is also evidenced,
implying the coupling between magnetism and ferroelectricity. Furthermore,
strong visible light absorption is observed, which can be understood from its
electronic structure. Its direct and appropriate band gap, as well as wide
conducting bands, makes Ca$_3$Mn$_2$O$_7$ a potential candidate for
ferroelectric photoelectric applications.
