Past exam of the options II course of the University of Oxford / 2008 printed variant / P / P4 / a / Solution

A QR factorization is $A=QR$, where $Q^{T}Q=I$ and $R$ is upper triangular. If $A$ is orthogonal, so is $R=Q^{T}A$. Its inverse is upper triangular, but $R^{-1}=R^T$ is lower triangular, forcing $R$ to be diagonal. Orthogonality then givesWith the customary positive diagonal convention, $R=I$.