If matrices $A$ and $B$ are similar (i.e., $B = P^{-1}AP$ for some invertible matrix $P$), which of the following must necessarily be true?

I. $A$ and $B$ have the same eigenvalues II. $A$ and $B$ have the same eigenvectors III. $\text{tr}(A) = \text{tr}(B)$ (equal traces) IV. $\det(A) = \det(B)$ (equal determinants)