There is a nondiagonal matrix $A\in M_3(\mathbb{R})$ s.t. $ I_{3} +A+A^2=O_3$?

Question

There is a nondiagonal matrix $A\in M_3(\mathbb{R})$ s.t. $ I_{3} +A+A^2=O_3$?

This question is related to this question . There exists a nondiagonal matrix $A\in \mathcal {M}_n (\mathbb{R}) $ s.t. $A^{k+1}=I_n $ and $I_n-A $ invertible?

Answer 1

There is no such real $3 \times 3$ matrix (diagonal or otherwise). In particular, we note that every $3 \times 3$ matrix with real entries has a real eigenvalue, but if $v$ is an eigenvector associated with real eigenvalue $\lambda$, then $$ (I + A + A^2)v = (1 + \lambda + \lambda^2)v \neq 0 $$