IMC 2025 Problem 8

For an $n \times n$ real matrix $A \in M_n(\mathbb{R})$ , denote by $A^{\mathbb{R}}$ its counter-clockwise $90^{\circ}$ rotation. (10 points) For example, $$\left[\begin{array}{lll} 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 \end{array}\right]^R=\left[\begin{array}{lll} 3 & 6 & 9 2 & 5 & 8 1 & 4 & 7 \end{array}\right]$$ Prove that if $A=A^R$ then for any eigenvalue $\lambda$ of $A$ , we have $\operatorname{Re} \lambda=0$ or $\operatorname{Im} \lambda=0$ .