Let $A\in\mathbb{R}^{n\times n}$ such that $A^2=-I$. Find all eigenvalues of A.

Question

The conclusion I got was A has no real eigenvalues.

Can anyone tell me if it's right or what are these eigenvalues?

Answer 1

If $Av = \lambda v$ then $$ A^2v = A( \lambda v) = \lambda (Av) = \lambda^2v $$

But $$A^2v = -Iv=-v \implies ( \lambda^2+1)v = 0$$

So $ \lambda =\pm i$.