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Is it possible to have a non-identity $2 \times 2$ diagonalizable, invertible, complex matrix $A$ s.t characteristics polynomials of $A$ and $A^2$ are the same?

I am not getting any hint even how to create one.

I can start with two different eigenvalues but for this, we won't have the same characteristic poly.

I was also trying to play with $$ \begin{pmatrix} 0 & i \\ i & 0 \\ \end{pmatrix} $$ Does not help.