 Solution

Complex conjugation permutes the irreducible characters, because the conjugate of an irreducible representation is irreducible. This involution fixes the real-valued characters and interchanges the other characters in pairs. If there are $r$ such pairs, then conjugating $A$ permutes its rows by $r$ transpositions. ThereforeIt follows that the determinant is real when $r$ is even and purely imaginary when $r$ is odd. It never vanishes, by the preceding part. Real-valued characters include both real and quaternionic representation types; the criterion concerns values of the characters, not those two representation types separately.