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We propose a modification of the usual algorithms for Gaussian elimination, determination of minors and computation of determinants in case of matrices with symbolic data (polynomials, rational functions, general algebraic structures and the like). The proposed modification improves on the Sasaki-Murao algorithm. The new algorithm aims at improving efficiency for a general class algebraic data like polynomials with non exact coefficients, algebraic structures with zero divisors, rational functions and general functions in several variables. The main difference to other approaches is that we replace the diagonal elements not at the beginning of the algorithm but then when they are taken as a pivot elements (independent wether or not these elements are non-zero). Therefore, our method abolishes the steps where pivot elements are checked to be non-zero, and even zero elements can be used as pivots.