Related to an answer to a previous question. The answer assume the following result:

Let $G$ be a finite group and $\rho : G \rightarrow \text{GL}(\mathbb{C}, n)$ be a faithful representation of $G$ (ie. $\text{Ker}(\rho) = 1_G$). Let $\chi$ be the character associated to $\rho$. Then, for all $g \in G$ such that $g \not= 1_G$ we have $|\chi(g)| < n$.

Is this true? If yes, why? I couldn't find any proof and I can't understand the small justification given in the previous answer.

Related to an answer to a previous question. The answer assume the following result:

Let $G$ be a finite group and $\rho : G \rightarrow \text{GL}(\mathbb{C}, n)$ be a faithful representation of $G$ (ie. $\text{Ker}(\rho) = 1_G$). Let $\chi$ be the character associated to $\rho$. Then, for all $g \in G$ such that $g \not= 1_G$ we have $|\chi(g)| < n$.

Is this true? If yes, why? I couldn't find any proof and I can't understand the small justification given in the previous answer.