A simple and systematic procedure for obtaining the determinant of any n x n matrix without the use of cofactors is developed in this publication. The procedure involves reduction of the square matrix to echelon form and dividing the product of the diagonal elements by the product of some numbers defined as lower multipliers. The simplicity of the procedure is maintained and appreciated as the order of the matrix n, tends to infinity. This makes it superior to the traditional method of cofactors which is often tedious to apply in high order matrices.