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Unformatted text preview: 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 2 8 10 1 3 2 2 7 1 2 1 1 3 2 1 4 5 1 3 2 2 7 + row 2 +2 row 2 1 1 3 2 1 4 5 1 7 1 10  row 3 1 1 3 2 1 4 5 1 7 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 . 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...
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 Spring '08
 alndy
 Determinant, Matrices

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