In this paper we prove the well known determinant a l representation of the Moore-Penrose generalized inverse of matrices, introduced by Moore, Arghiriade, Dragomir and Gabriel on an elementary way. It is shown that determinantal representation of $\{1,2,3\}$, $\{1,2,4\}$ and $\{1,2\}$ inverses can be obtained in a similar way. Finally, we present a short proof for the determinantal representation of the Moore--Penrosc solution of a system of linear equations.