Showing det(X) = det(Y)

Question

Using concepts of diagonalization where $X = P Y P^{-1}$, I have to show that $\det(X) = \det(Y)$.
I do understand that $PP^{-1} = I$ and since $\det(I) = 1$.
So how would I correctly explain that both these statements help prove the given question?

Answer 1

We have the rule $\det(AB) = \det(A)\det(B)$. Since $X = PYP^{-1}$ we have $\det(X) = \det(PYP^{-1}) = \det(P) \det(Y) \det(P^{-1}) = \frac{\det(P)}{\det(P)} \det(Y) = \det(Y)$. Since $\det(P^{-1}) = \frac{1}{\det(P)}$