Prove that if $AA^T=A^TA$ and $AB=BA$ then $AB^T=B^TA$

Question

Prove that if $AA^T=A^TA$ and $AB=BA$ then $AB^T=B^TA$ where $A$ and $B$ are matrices.

It doesn't say of what order they are. Can somebody help me with this exam problem?

I know I should put my work here but I really don't know what to do. I would really like some help.

Answer 1

Remarks:

1) $AB = B A \Rightarrow B^T A^T = A^T B^T$;

2) $A B^T = B^T A \iff B A^T = A^T B \iff B A^T (AB) = A^T B (AB) $,