Computation of determinant

where Sn (also known as the symmetric group on n elements) is the set of all permutations of {1, 2, . . . , n}, i.e., all the ways of pairing up the n rows of the matrix with the n columns, and I(π) is the inversion number of π, the minimal number of transpositions of adjacent columns needed to turn π into the identity permutation. This formula [1] is practical for some simple matrices such as 3-by-3 and 4-by-4 matrices, but for large dense matrices it is inefficient, since there have many of computation. At present, most mathematicians are familiar with Gaussian elimination as a more practical method of evaluating determinants, while there have many of filled elements for a sparse matrix. Problem: The problem is how to deal with the algebraic expression (1) to reduce its computation and at the same time increase its accuracy for some special matrices.