$C-2I = (-1)^n \cdot 7^m$

Question

I am stuck on this linear algebra question. Please help!
Let C be a real $n \times n$ matrix such that $C^3 = I$

(i) What are the possible eigenvalues of C?

(ii) Show that $$\det(C-2I) = (-1)^n \cdot7^m$$ For some integer $m$ with $0 \leq m\leq n/2.$

I know how to do section i but added it in for the potential context.

Answer 1

Hint

$(i)$ If $\lambda$ is a eigenvalue of $C$ and $C^3=I$ then $\lambda^3=1$

$(ii)$ $$C-2I=K\to C=K+2I$$ $$C^3=K^3+6K^2+12K+8I$$ $$K^3+6K^2+12K+7I=0$$

Now look the eigenvalues of $K$ and use that $\det K=\text{product of eigenvalues}$.