If $A$ is a $2 × 2$ complex matrix that is invertible and diagonalizable, and such that $A$
and $A^2$ have the same characteristic polynomial, then $A$ is the $2 × 2$ identity matrix.

My claim:
Eigenvalues of $A^2$ are square of eigenvalue of $A$

$$\lambda=\lambda^2$$Since invertible $\lambda=1$ hence similar to identity matrix. 

But only matrix similar to identity is identity itself.

But answer is given as FALSE. please explain me why i'm wrong