Let $G$ be a finite group and $\left\{\chi_{i}\right\}$ the set of its irreducible characters. Also choose representatives $g_{j}$ for the conjugacy classes, and denote by $Z\left(g_{j}\right)$ their centralisers.

(i) State the orthogonality and completeness relations for the $\chi_{k}$.

(ii) Using Part (i), or otherwise, show that

$$\sum_{i} \overline{\chi_{i}\left(g_{j}\right)} \cdot \chi_{i}\left(g_{k}\right)=\delta_{j k} \cdot\left|Z\left(g_{j}\right)\right|$$

(iii) Let $A$ be the matrix with $A_{i j}=\chi_{i}\left(g_{j}\right)$. Prove that

$$|\operatorname{det} A|^{2}=\prod_{j}\left|Z\left(g_{j}\right)\right|$$

(iv) Show that $\operatorname{det} A$ is either real or purely imaginary, explaining when each situation occurs.

[Hint for (iv): Consider the effect of complex conjugation on the rows of the matrix A.]