Matrix A satisfy in $A^n=2A$, Prove it is diagonalizable

Question

Let $A$ is a $n\times n$ matrix for $n>1$ with complex entries that satisfy in $A^n=2A$. Prove that $A$ is a diagonalizable matrix.

Answer 1

Since $A^n - 2A = 0$, then we know that the minimal polynomial $m \mid x^n - 2x = x(x^{n-1} - 2)$. In particular, this says that the roots of the minimal polynomial comprises of a subset of the roots of the latter polynomial, which you can easily identify to be a separable polynomial.

Now, there is a theorem using Jordan Canonical Forms that says

A matrix is diagonalizable over a suitable field (such as $\mathbf{C}$) if and only if its minimal polynomial is separable

Answer 2

Apply the Cayley-Hamilton theorem.