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 ³.´/a 1;±.1/ ²²² a n;±.n/ the summand belonging to ´ is zero unless ´.i/ ³ i for all i . But the only permutation ´ with this property is the identity permutation. Therefore det .A/ D a 1;1 a 2;2 ²²² a n;n : To prove (d), ﬁx 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
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 Algebra, Determinant, Matrices, Invertible matrix, Diagonal matrix, det.A/, Matn .R/

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