$A$ is a non singular $n \times n$ matrix with all eigenvalues real. The trace $T(A^2)= T(A^3)= T(A^4)$. Find $T(A)$.

I have tried using the fact that trace is a linear functional and tried to explore its kernel, but have made no progress. Can I get some help?

My knowledge of linear algebra is limited to the first $5$ chapters of Hoffman and Kunze, basic properties of eigenvalues, annihilating polynomials and invariant subspaces. Is it sufficient to do this problem?