# Anticommuting matricies

• May 1st 2009, 08:14 AM
cubs3205
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?
• May 1st 2009, 08:49 AM
Opalg
Quote:

Originally Posted by cubs3205
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.