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240assign10 - MAT240 Assignment 10 Partial Solutions Arthur...

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MAT240 Assignment 10 — Partial Solutions Arthur Fischer April 1, 2004 Exercise (#17, p.229). Let A,B M n × n ( F ) be such that AB = - BA . Prove that if n is odd and F is not a field of characteristic two, then A or B is not invertible. Solution. Assume as in the statement of the problem that F is not a field of characteristic two, that n is odd, and that A,B M n × n ( F ) are such that AB = - BA . By various theorems and previous exercises, we have det( A ) · det( B ) = det( AB ) = det( - BA ) = ( - 1) n det( BA ) = - det( B ) · det( A ) . Since F is not a field of characteristic two, the only field-element which is its own negative is 0, and therefore det( A )det( B ) = 0. By properties of fields, this only occurs when either det( A ) = 0 or det( B ) = 0, immediately implying that either A or B is not invertible. Exercise (#6, p.238). Prove that if M M n × n ( F ) can be written in the form M = ± A B 0 C ² , where A and C are square matrices, then det( M ) = det( A ) · det(
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240assign10 - MAT240 Assignment 10 Partial Solutions Arthur...

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