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  • In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact. The method can also be used to compute the determinant of matrices with real entries, avoiding the introduction any round-off errors beyond those already present in the input. During the execution of Bareiss algorithm, every integer that is computed is the determinant of a submatrix of the input matrix. This allows, using Hadamard inequality, to bound the size of these integers. Otherwise, Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic operations. It follows that, for an n × n matrix of maximum value 2ᴸ for each entry, the Bareiss algorithm runs in O elementary operations with an O bound on the absolute value of intermediate values needed. Its computational complexity is thus O when using elementary arithmetic or O by using fast multiplication.