Unformatted text preview: R is a ﬁeld. One can reduce a given matrix A to triangular form by elementary row operations: interchanging two rows or adding a multiple of one row to another row. Operations of the ﬁrst type change the sign of the determinant while operations of the second type leave the determinant unchanged. If B is an upper triangular matrix obtained from A in this manner, then det .A/ D . ± 1/ k det .B/ , where k is the number of row interchanges performed in the reduction. But det .B/ is the product of the diagonal entries of B , by part (c) of Corollary 8.3.9 . The same method works for matrices over an integral domain, as one can work in the ﬁeld of fractions; of course, the determinant in the ﬁeld of fractions is the same as the determinant in the integral domain....
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 Fall '08
 EVERAGE
 Algebra, Determinant, Characteristic polynomial, 2 m, multilinearity

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