College Algebra Exam Review 352

College Algebra Exam Review 352 - A is lower triangular;...

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362 8. MODULES (ii) det .E n / D 1 , where E n is the n –by– n identity matrix. (b) If ± W Mat n .R/ ±! N is any function that, regarded as a func- tion on the columns of a matrix, is alternating and multilinear, then ±.A/ D det .A/±.E n / for all A 2 Mat n .R/ . Proof. This follows immediately from the properties of ² given in Propo- sition 8.3.6 . n Corollary 8.3.9. Let A and B be n –by– n matrices over R . The determi- nant has the following properties (a) det .A t / D det .A/ , where A t denotes the transpose of A . (b) det .A/ is an alternating multilinear function of the rows of A . (c) If A is a triangular matrix (i.e. all the entries above (or below) the main diagonal are zero) then det .A/ is the product of the diagonal entries of A . (d) det .AB/ D det .A/ det .B/ (e) If A is invertible in Mat n .R/ , then det .A/ is a unit in R , and det .A ± 1 / D det .A/ ± 1 . Proof. The identity det .A t / D det .A/ of part (a) follows from the equality of the two formulas for ² in Proposition 8.3.6 . Statement (b) follows from (a) and the properties of det as a function on the columns of a matrix. For (c), suppose that
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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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