Solutions4 - Math 601 Solutions to Homework 4 1 Use row...

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Math 601 Solutions to Homework 4 1. Use row reduction to compute the determinants of the following ma- trices: (a) A = 1 1 3 2 1 3 11 12 2 1 3 6 4 2 5 8 (b) B = a a a a a a b 2 b b b b b c c 3 c c c c d d d 4 d d d e e e e 5 e e 1 1 1 1 1 6 Answer: (a) We row reduce the matrix: 1 1 3 2 1 3 11 12 2 1 3 6 4 2 5 8 - row 1 - 2 row 1 - 4 row 1 -→ 1 1 3 2 0 2 8 10 0 - 1 - 3 2 0 - 2 - 7 0 × 1 2 1 1 3 2 0 1 4 5 0 - 1 - 3 2 0 - 2 - 7 0 + row 2 +2 row 2 -→ 1 1 3 2 0 1 4 5 0 0 1 7 0 0 1 10 - row 3 -→ 1 1 3 2 0 1 4 5 0 0 1 7 0 0 0 3 The determinant of the row reduced matrix is 3 (because the ma- trix is triangular, so the determinant is the product of the di- agonal entries). Most of the row operations did not change the determinant. When we multiplied the second row by 1 2 , we also multiplied the determinant by 1 2 . This is the only row operation that changed the determinant. Thus, the original matrix, had determinant 2 · 3 = 6. Thus, det( A ) = 6 (b) We can begin by dividing the first row by a , the second row by b , the third row by c , the fourth row by d , and the fifth row by e .
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We get: fl fl fl fl fl fl fl fl fl fl fl fl a a a a a a b 2 b b b b b c c 3 c c c c d d d 4 d d d e e e e 5 e e 1 1 1 1 1 6 fl fl fl fl fl fl fl fl fl fl fl fl = abcde fl fl fl fl fl fl fl fl fl fl fl fl 1 1 1 1 1 1 1 2 1 1 1 1 1 1 3 1 1 1 1 1 1 4 1 1 1 1 1 1 5 1 1 1 1 1 1 6 fl fl fl fl fl fl fl fl fl fl fl fl Now, we can subtract the first row from each of the other rows: fl fl fl fl
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