Can anybody help me with these questions as I am completely lost and haven't done any character theory or rep theory for a long time
(1) Let X be the matrix comprising the character table of a group.
Show that X is non-singular & hence prove that the number of real-valued characters is equal to the number
of self-inverse conjugacy classes.
(2) Show that in a group G of odd order, no element other than the identity
is conjugate to its inverse. Deduce that in a group of odd order, the trivial character is the only
irreducible character that is real-valued.