I need help with this. So I'm working with $-I$ with $nxn$ size, so computing the eigenvalues for $n=1,2,3...$ gives me $-1$ as the unique eigenvalue for that matrix with $multiplicity=n$. Should I translate that into $A^2 = -I$?
The other bit of advice our teacher gave us was that when proven, $n$ had to be necessarily an even number.
Thank you in advance!
As mentioned by @angryavian in the comment, the antisymmetry is not necessary here. Let $\lambda$ an eigenvalue of $A$ with eigenvector $v\neq 0$, i.e., $Av=\lambda v$. Then $$-v = A^2 v = A(\lambda v) = \lambda A v = \lambda^2 v. $$ Hence, $\lambda^2 = -1$.