Since similarity of matrices' is an equivalence relation, doesn't that imply that given any polynomial equation involving similar matrices you can substitute in any similar matrices' and the equation will still hold?
For example, given $A,B,C\in M^F_{n\times n}$
if $B \cong C$ then: $A \cong B^2+5B+3I \iff A \cong C^2+5C+3I$
Correct?