Which one is not always true: $\det(A^tB)=0$ or $\det(AB^t) = 0$?

Question

I had this on an exam recently and I am not exactly sure about it.

Let $A,\ B$ be matrices with $A,\ B \in M_{m \times n}(K)$ with $m < n$ and $K$ a field. Which of the following statements is not always true? $$\det(A^tB)=0$$ $$\det(AB^t) = 0$$

Which is the one and why?

Answer 1

Hint: $A$ and $B$ have rank at most $m$. So, both $A^TB$ and $AB^T$ have rank at most $m$. How does that relate to the determinant?