= 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. Therefore
$$
\overline{\det A}=(-1)^r\det A.
$$
It follows that \b[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.
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