If I know that $\chi$ is a character of degree $2$ on $G$ and suppose $\chi(g) = 1$. Prove that $\chi(g^2)= -1$.
Now, I know that $\rho(g)$ is a 2x2 matrix, so therefore there are 2 eigenvalues (which could be the same).
And $\chi(g) = Tr(\rho(g)) = \lambda_1 + \lambda_2 = 1$
I also know that $\lambda_1 $and $\lambda_2$ are the n-th roots of unity.
My question is, how do I use this information to prove that $\lambda_1^2 + \lambda_2^2 = -1$?