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College Algebra Exam Review 357

College Algebra Exam Review 357 - 367 8.3 MULTILINEAR MAPS...

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8.3. MULTILINEAR MAPS AND DETERMINANTS 367 Example 8.3.17. An element of Mat n . Z / has an inverse in Mat n . Q / if its determinant is nonzero. It has an inverse in Mat n . Z / if, and only if, its determinant is ˙ 1 . Permanence of identitities Example 8.3.18. For any be an n –by– n matrix, let ˛.A/ denote the trans- pose of the matrix of cofactors of A . I claim that (a) det .˛.A// D det .A/ n 1 , and (b) ˛.˛.A// D det .A/ n 2 A . Both statements are easy to obtain under the additional assumption that R is an integral domain and det .A/ is nonzero. Start with the equation A˛.A/ D det .A/E , and take determinants to get det .A/ det .˛.A// D det .A/ n . Assuming that R is an integral domain and det .A/ is nonzero, we can cancel det .A/ to get the first assertion. Now we have ˛.A/˛.˛.A// D det .˛.A//E D det .A/ n 1 E; as well as ˛.A/A D det .A/E . It follows that ˛.A/ ˛.˛.A// det .A/ n 2 A D 0: Since det .A/ is assumed to be nonzero, ˛.A/ is invertible in Mat n .F / , where F is the field of fractions of R . Multiplying by the inverse of ˛.A/ gives the second assertion.
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