Given that $A^3=-A$ show that $A$ is not invertible.

Question

Let $A$ be an arbitrary square matrix with real numbers as elements. Given that $A^3=-A$ show that $A$ is not invertible.

This question appeared in my linear algebra book in the chapter on determinants so I assume that I'm supposed to show that $\det (A)=0$. I don't know how to do this, some help would be greatly appreciated.

Answer 1

For odd $n$ we have $\det(A)^3=(-1^n)\det(A)$. Which for nonzero $\det(A)$ gives $\det(A)^2=-1$ which is impossible for a real matrix.