OIMU 2005 Problem 2

Let $A,B,C$ be real square matrices of order $n$ such that $A^3=-I$ , $BA^2+BA=C^6+C+I$ and $C$ is symmetric. Is it possible that $n=2005$ ?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.