Determinants of grids, tori, cylinders and Mِbius ladders

Recently, Bieñ [A. Bieñ, The problem of singularity for planar grids, Discrete Math. 311 (2011) 921–931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box products of paths, Discrete Math. 312 (2012) 1844–1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, that of a cylinder, and that of a Möbius ladder.