Transpose and Antisymmetric Matrices

Question

I am trying to prove the following:

Given two complex matrices $A,B$ such that $AB = -BA$, prove that $(AB)^{T} = -AB$. Is this possible or should I make more assumptions?.

Thanks.

Answer 1

Transpose what you know : $(AB)^T = -A^TB^T$

Now if the claim is to hold, the following must hold:

$$-A^TB^T = -AB$$

This is true for example if $A^T=A,B^T=B$ but that is far from always the case.

It can also be true if $(AB)^{-1}(A^TB^T) = I$

which could be true for example if $A^{-1}A^T=I, B^{-1}B^T=I$ which gives the same conclusion but the inverses must exist.

So if additionally $A^T=A,B^T=B$ then it is true.

Note that for complex matrices conjugate transpose $^*$ is much more common than $^T$