VTRMC 1979 Problem 3

Let $A$ be an $n\times n$ nonsingular matrix with complex elements, and let $\overline{A}$ be its complex conjugate. Let $B = A\overline{A}+I$ , where $I$ is the $n\times n$ identity matrix. (a) Prove or disprove: $A^{-1}BA = \overline{B}$ . (b) Prove or disprove: the determinant of $A\overline{A}+I$ is real.