A and its transpose have the same minimal polynomial

Question

Help, I need to prove the next

Prove that A and A^t have the same minimal polynomial

I have proved that they have the same characteristic polynomial, therefore its minimal polinomial divides the characteristic poynomial, but, is that enough to prove it?

Answer 1

Hint: Go back to the definition of the minimal polynomial noting that for every polynomial $p$, we have $p(A^T) = p(A)^T$.