Erwin H. Bareiss developed integer-preserving methods for Gaussian elimination. The Bareiss algorithm uses determinant identities to arrange divisions that are exact, reducing the growth of intermediate rational expressions.[ref]
An update combines two products and divides by a previous pivot. The reason this stays in the integers is structural: the quantities represent related minors satisfying Sylvester identity for bordered minors. This makes exact linear algebra practical without replacing the calculation by floating-point approximation.[ref]
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