Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 6C / b

If $B=S^{-1}AS$, thenso the characteristic polynomials, eigenvalues, and algebraic multiplicities agree. The converse is false: $I_2$ and $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ have the same characteristic polynomial, but the first is diagonalisable and the second is not, so they cannot be similar.