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Help with a linear algebra proof

  1. Mar 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Let A and B be n x n matrices.
    Show that if AB = -BA and n is odd, then A or B is singular.

    2. Relevant equations

    - A matrix is singular iff its determinant is 0.
    or possibly: Theorem: if A and B are both n x n matrices, then |AB| = |A||B|

    3. The attempt at a solution

    I kind of have a proof, but it doesn't seem correct.

    AB = -BA
    therefore
    |A||B| = -(|B||A|)
    2|A||B| = 0
    |A||B| = 0
    therefore |A| = 0 or |B| = 0
    thus, A or B is singular.

    This doesn't make use of the fact that n is odd, as specified, which is why I don't think it's correct.
     
  2. jcsd
  3. Mar 26, 2009 #2
    Remember that, if k is a scalar and A is an n x n matrix, then |kA|=k^n*|A|, not k*|A|. In this case, your proof is correct, but only because the matrix has an odd dimension, as that allows you to "pull out" the negative sign
     
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