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Unformatted text preview: A is lower triangular; that is the matrix entries a i;j are zero if j > i . In the expression det .A/ D X 2 S n ./a 1;.1/ a n;.n/ the summand belonging to is zero unless .i/ i for all i . But the only permutation with this property is the identity permutation. Therefore det .A/ D a 1;1 a 2;2 a n;n : To prove (d), x a matrix A and consider the function W B 7! det .AB/ . Since the columns of AB are Ab 1 ;:::;Ab n , where b j is the j th column of B , it follows that is an alternating mulilinear function of the columns of B . Moreover, .E n / D det .A/ . Therefore det .AB/ D .B/ D det .A/ det .B/ , by part (b) of the previous corollary. If A is invertible, then 1 D det .E n / D det .AA 1 / D det .A/ det .A 1 /; so det .A/ is a unit in R , and det .A/ 1 D det .A 1 / . n...
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 Fall '08
 EVERAGE
 Algebra

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