h10 - (Hint: The formula for det( A ) is similar to the...

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Homework Set 10 for MAS 4105 Due Friday, March 30 1. Let A be an n × n matrix such that A t = - A (we say that A is skew-symmetric ). Prove that if n is odd then A is not invertible. 2. Determine (with proof) a formula for det( A ) in terms of det( ˜ A n 1 ), where A = a 11 a 12 . . . a 1 ,n - 1 a 1 n a 21 a 22 . . . a 2 ,n - 1 a 2 n . . . . . . . . . . . . a n - 1 , 1 a n - 1 , 2 . . . a n - 1 ,n - 1 a n - 1 ,n a n 1 0 . . . 0 0 3. Find a formula for det( A ), where A is given by A = a 11 a 12 . . . a 1 ,n - 1 a 1 n a 21 a 22 . . . a 2 ,n - 1 0 . . . . . . . . . . . . a n - 1 , 1 a n - 1 , 2 . . . 0 0 a n 1 0 . . . 0 0 . Use mathematical induction to prove that your formula is correct.
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Unformatted text preview: (Hint: The formula for det( A ) is similar to the formula for the deter-minant of an upper triangular matrix, but with a sign that depends on n .) The following problems are strongly recommended, but should not be turned in: 4.3: 1, 8, 9, 10, 11, 13, 14, 15 4.4: 1, 2ac, 3ace, 4aceg, 5, 6 5.1: 1, 2, 3, 4, 9, 10, 11, 12 5.2: 1, 2aceg, 3acde, 4, 7, 8, 9, 11, 12...
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