The determinant of a complex matrix A, denoted as det(A), satisfies the property det(A*) = (det(A))* where A* represents the complex conjugate of A. This relationship holds because the determinant is a polynomial with real coefficients derived from the entries of A. Specifically, for any polynomial p with real coefficients, the equality p(x_1*, x_2*, ..., x_n*) = p(x_1, x_2, ..., x_n)* is valid, confirming the stated property of determinants.
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Citan Uzuki said:Yes. The determinant is a polynomial with real coefficients of the entries in A, and for any polynomial p in n variables with real coefficients, p(x_1*, x_2*, ... , x_n*) = p(x_1, x_2, ... , x_n)*.