find the set of all positive integers n for which there is a real matrix A of dimension n×n such that $A^{−1}=−A$.

Question

Need to find the set of all positive integers $n$ for which there is a real matrix $A$ of dimension $n\times n$ such that $A^{−1}=−A$.

Tried: let $\lambda$ be an eigen value of A then we have $\lambda + \lambda^{-1}=0$ $\implies$ $\lambda^{2}=-1$$\implies$ $\lambda=\pm i$ now I am not getting how I do it.

Answer 1

The characteristic polynomial has no real roots, so it must be of even degree. Hence $n$ is even. To finish, construct a square matrix of any even size that squares to $-I$.