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

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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 0 1 + x 2 x 3 · · · x k +1 0 x 2 1 + x 3 · · · x k +1 . . . . . . . . . . . . . . . 0 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 . . . . . . . . . . . . . . . x 1 x 2 x 3 · · · 1 + x k +1 . By the induction hypothesis, the first term on the right-hand side is equal to 1 + x 2 + · · · + x k +1 . To evaluate the second term, we subtract the first row of the matrix from each of the other rows, getting det

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

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