[Math] Characteristic Polynomial of a matrix and its transpose

Question

Does a matrix and and its transpose have the same characteristic polynomial? I know that they have the same eigenvalues but different eigenvectors. Does having the same eigenvalues mean they share the same characteristic polynomial?

Answer 1

For all $A \in \mathcal{M}_n(\mathbb{K})$, $$\det(A-XI_n)=\det(^t(A-XI_n))=\det(^tA - XI_n)$$