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Math Help - Anticommuting matricies

  1. #1
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    Anticommuting matricies

    I know that two matrices anticommute when AB = -BA
    However, apparently for n x n matrices when n is odd, either A or B is not invertible. But when n is even, both can be invertible
    Why is this the case?
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  2. #2
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    Quote Originally Posted by cubs3205 View Post
    I know that two matrices anticommute when AB = -BA
    However, apparently for n x n matrices when n is odd, either A or B is not invertible. But when n is even, both can be invertible
    Why is this the case?
    Look at the determinants: \det(-BA) = (-1)^n\det(BA) = (-1)^n\det(AB). If AB=–BA, it follows that either n is even or det(AB)=0.
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