Is $A$ the $2 × 2$ identity matrix

Question

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

Answer 1

Counterexample: $$A=\begin{pmatrix}e^{\frac{2}{3}\pi i} & 0\\ 0 & e^{\frac{4}{3}\pi i}\end{pmatrix}$$ and $$A^{2}=\begin{pmatrix}e^{\frac{4}{3}\pi i} & 0\\ 0 & e^{\frac{2}{3}\pi i}\end{pmatrix}.$$