# sum of elements of an irreducible representation of a group

Show that the sum over a group of the matrix elements of any irreducible representation other than the identity representation is equal to zero.

This is the exercise in a group theory textbook, the question is the exactly as the book presents. Thanks for help.

For all $h$ in $G$, $$\sum_{g{\rm\ in\ }G}\rho(g)=\sum_{g{\rm\ in\ }G}\rho(gh)=\sum_{g{\rm\ in\ }G}\rho(g)\rho(h)=\left(\sum_{g{\rm\ in\ }G}\rho(g)\right)\rho(h)$$ so it reduces to showing that $I-\rho(h)$ is invertible for some $h$ in $G$.
• Thanks. But if for all h in G, $I-\rho(h)$ is not invertible, how to deduce a contradiction? – yangcs11 May 20 '18 at 11:48