If matrix $A$ is similar to matrix $D$ and $B$ is similar to $E$, than: $AB$ is similar to $DE$?

Question

More specifically: if $A$ & $B$ are diagonalizeable, than is it correct to say that $AB$ is diagonalizeable? (Hints would be more appreciated)

Answer 1

Hint: Try to see what happens in a concrete example where $A,B$ are diagonalizable but $AB \neq BA$ (so $A$ and $B$ don't commute).

Answer 2

we have: $$ A=P^{-1}DP \quad \land \quad B=Q^{-1}EQ \quad \Rightarrow \quad AB=P^{-1}DPQ^{-1}EQ $$ So we can say that if $P=Q$ ( i.e. $A$ and $B$ are simultaneously diagonalizable) than $AB$ is diagonalizable.