= Solution
Let $\rho$ afford the <character> $\chi$, and let $m$ be the <order of a group element> $g$. Then $\rho(g)^m=I$, so its <minimal polynomial> divides $T^m-1$. Over the <complex numbers> this <polynomial> has distinct <polynomial roots>, since its <formal derivative> $mT^{m-1}$ does not vanish at any of its <polynomial roots>. Thus $\rho(g)$ is <diagonalizable>, with <eigenvalues> $\lambda_1,\ldots,\lambda_d$ that are <roots of unity>.
Each $\lambda_j$ has <absolute value> one, so $\lambda_j^{-1}=\overline{\lambda_j}$. The <eigenvalues> of $\rho(g^{-1})=\rho(g)^{-1}$ are $\lambda_j^{-1}$, giving
$$
\boxed{\chi(g^{-1})=\sum_j\lambda_j^{-1}
=\sum_j\overline{\lambda_j}
=\overline{\chi(g)}.}
$$
This is the <character inversion identity>.
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