A4S - ASSIGNMENT 4 Solutions A matrix is skew-symmetric if...

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Unformatted text preview: ASSIGNMENT 4 Solutions A matrix is skew-symmetric if A T =- A . Let A be an n n skew-symmetric matrix, with n odd. 1. [3 marks] Prove that A is not invertible. We have det ( A T ) = det A = det (- A ) . Let E i denote the elementary matrix corresponding to the elementary row operation that scales row i by- 1 . Then det (- A ) = det ( E 1 E n A ) = det E 1 det E n det A = (- 1) n det A ; that is, det A = (- 1) n det A. If n is odd, then det A = 0 , hence A is not invertible. Let I be the n n identity matrix, and A be the following n n matrix: A = x 1 x 2 x n x 1 x 2 x n . . . . . . . . . x 1 x 2 x n . 2. [4 marks] Prove that det ( I + A ) = 1 + x 1 + + x n . We show this by induction on n . The n = 2 case is clear: we have det 1 + x 1 x 2 x 1 1 + x 2 = (1 + x 1 ) (1 + x 2 )- x 1 x 2 = 1 + x 1 + x 2 . We assume the claim holds for n = k . Now det ( I + A ) = det 1 x 2 x 3 x k +1 1 + x 2 x 3 x k +1 x 2 1 + x 3 x k +1 . . . . . . . . . . . . . . . x 2 x 3 1 + x k +1 + det x 1 x 2 x 3 x k +1 x 1 1 + x 2 x 3 x k +1 x 1 x 2 1 + x 3 x k +1 . . . . . . . ....
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This note was uploaded on 04/07/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.

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A4S - ASSIGNMENT 4 Solutions A matrix is skew-symmetric if...

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