# Math Help - If A^2 = -I then prove...

1. ## If A^2 = -I then prove...

Given an n x n matrix A with real entries such that A^2 = -I, prove the following about A:

1) n is even
2) A has no real eigenvalues
3) det A = 1

2. Hints :

1) $(\det A)^2=(-1)^n$

2) If $\lambda \in\mathbb{R}$ is an eigenvalue of $A$ , then $Ax=\lambda x$ for some $0\neq x\in \mathbb{R}^n$ .

Prove that $(\lambda^2+1)x=0$ (contradiction).

3) No hint, try it.

Fernando Revilla