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.