Let B be any real, nonsingular $nxn$ matrix, where n is even, and set $A=B-B^T$. Show that A does not admit an LU decomposition without pivoting.
I know that A is a skew symmetric matrix. The 2x2 case is
It is easy to see we can't eliminate a so we must pivot. What I don't understand is why n must be even? I think even odd cases show this... Thanks.