Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 4 / i

An orthogonal matrix satisfies $A^{T}A=I$. Taking determinants and using $\det A^T=\det A$ gives $(\det A{)}^2=1$, soThis calculation remains valid over the complex numbers: orthogonality uses the transpose, whereas a unitary matrix uses the conjugate transpose.