The similar matrix of a matrix $A$ is $P^{-1}AP$ where P is invertible.

The characteristic equation of $P^{-1}AP$ is $|P^{-1}AP - \lambda I| = 0$. Now,

$|P^{-1}AP - \lambda I| \\ = |P^{-1}AP - P^{-1}\lambda IP| \\ = |P^{-1}(A-\lambda I)P \\ = |P^{-1}||A-\lambda I||P| \\ = |A-\lambda I| = 0$

which is the characteristic equation of the matrix $A$. Hence, proved.