Enumeration of matrices with single unit-entry rows over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, formal expressions of the number of [Formula: see text] matrices over [Formula: see text] of rank [Formula: see text] and the number of invertible matrices over [Formula: see text] are presented. The number of matrices over [Formula: see text] with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined.