$\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^T$

Question

I want to prove that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^T$.

We know that $\det(A)=\det(A^T)$, but how can I apply it to $\det(A-\lambda I)$?

Thanks a lot!