Help with a linear algebra proof

[paulrb]

Homework Statement

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

Homework 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|

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.

 

[Wretchosoft]

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