IMC 2025 · Problem 8

For an $n \times n$ real matrix $A \in M_n(\mathbb{R})$, denote by $A^{\mathsf{R}}$ its counter-clockwise $90°$ rotation. For example,

$$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}^{\mathsf{R}} = \begin{bmatrix} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{bmatrix}.$$

Prove that if $A = A^{\mathsf{R}}$ then for any eigenvalue $\lambda$ of $A$, we have $\operatorname{Re}\lambda = 0$ or $\operatorname{Im}\lambda = 0$.