Let $A$ be a $2 \times 2$ complex matrix such that $A^2$ is diagonalizable and $det(A)\ne 0$. Prove that A is diagonalizable

Question

Let $A$ be a $2 \times 2$ complex matrix such that $A^2$ is diagonalizable and $det(A)\ne 0$. Prove that A is diagonalizable

I have no idea how to even begin the proof.

Answer 1

Hint: Any non-diagonalizable $2\times2$ complex matrix is similar to a matrix of the form $\left[\begin{smallmatrix}\alpha&\beta\\0&\alpha\end{smallmatrix}\right]$.