On the rank of incidence matrices for points and lines of finite affine and projective geometries over a field of four elements

We consider incidence matrices for points and lines of affine and projective geometries over a field of four elements. For such matrices we derive a simple formula for the $2$-rank and, as a consequence, new combinatorial identities expressing the relation of the obtained formulas for the rank with previously known formulas. We also present a way to construct generating systems for rows of these matrices.