A famous problem in linear algebra is that "A $n\times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$."

I know one proof using Smith Normal form. However, I want to know an elementary proof avoiding any concepts related to SNF.

My question is: "Is there an elementary way to prove this?"

Requirements are:

Do NOT use the structure theorem over PID.

Do NOT use the Smith Normal form (nor Jordan canonical form).

Do NOT use the concept of invariant factors.

Provide an explicit invertible matrix P such that $$A=PA^T P^{-1}.$$