Let A and B be similar nxn matrices. Prove that if A is idempotent, then B is idempotent. (X is idempotent if X^2=X )

Step 1
According to the given information, it Let A and B be $n\times n$ similar matrices.
If A is idempotent the show that B is idempotent.
$A\text{ is idempotent }\Rightarrow A^2=A$
$\text{ to show }B\text{ is idempotent }{B}^2=B$
Step 2
A and B are similar matrices so, there exist an invertible matrix P such that: $B=P^{-1}AP...(A)$
Step 3
Square both sides:
$B^2=(P^{-1}AP)(P^{-1}AP)$
$B^2=P^{-1}A(P{P}^{-1})AP[P{P}^{-1}=I]$
$B^2=P^{-1}A(I)AP$
$B^2=P^{-1}{A}^{2}P$
$B^2=P^{-1}AP[\text{by given condition }{A}^2=A]$
$B^2=B[\text{from equation }(A)]$
Therefore, B is also an idempotent matrix.