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College Algebra Exam Review 355

College Algebra Exam Review 355 - ± 1 i C j a i;j det.A...

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8.3. MULTILINEAR MAPS AND DETERMINANTS 365 Proposition 8.3.14. (Cofactor Expansion) Let A be an n –by– n matrix over R . (a) For any i , det .A/ D n X j D 1 . ± 1/ i C j a i;j det .A i;j /: (b) If i ¤ k , then 0 D n X j D 1 . ± 1/ i C j a k;j det .A i;j /: Proof. Fix i and j Let B j be the matrix matrix obtained from A by replac- ing all the entries of the i –th row by 0 ’s, except for the entry a i;j , which is retained. Perform i C j ± 2 row and column interchanges to move the entry a i;j into the .1;1/ position. The resulting matrix is B 0 j D 2 6 6 6 6 6 4 a i;j 0 ²²² 0 a 1;j a 2;j : : : a n;j A i;j 3 7 7 7 7 7 5 : That is, a i;j occupies the .1;1/ position, the remainder of the first row is zero, the remainder of the first column contains the remaining entries from the j –th column of A , and the rest of the matrix is the square matrix A i;j . According to Lemma 8.3.13 , det .B 0 j / D a i;j det .A i;j / . Therefore det .B j / D . ± 1/ i C j det .B 0 j / D
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Unformatted text preview: . ± 1/ i C j a i;j det .A i;j /: Since the matrices B j are identical with A except in the i –th row, and the sum of the i –th rows of the B j ’s is the i –th row of A , we have det .A/ D X j det .B j / D n X j D 1 . ± 1/ i C j a i;j det .A i;j /: This proves (a). For (b), let B be the matrix that is identical to A , except that the i –th row is replaced by the k –th row of A . Since B has two identical rows, det .B/ D . Because B is the same as A except in the i –th row, B i;j D A i;j for all j . Moreover, b i;j D a k;j . Thus, D det .B/ D X j . ± 1/ i C j b i;j B i;j D X j . ± 1/ i C j a k;j A i;j :...
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