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