(Solved): 11. (i) If \( U \) is a unitary matrix, show that \( |\operatorname{det} U|=1 \). (Hint: remember ...

11. (i) If \( U \) is a unitary matrix, show that \( |\operatorname{det} U|=1 \). (Hint: remember \( \bar{U}^{\mathrm{T}} U=\mathrm{I}_{n} \), and take determinants of both sides, noting that \( \operatorname{det}\left(\bar{U}^{T}\right)=\overline{\operatorname{det} U} \). Why?) It follows that a real unitary matrix has determinant equal to \( \pm 1 \). The sign determines whether it is a rotation or a reflection matrix.