Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 17A / c / Solution

At step $k$, choose a Householder reflection acting only on coordinates $k,\dots ,n$ that maps the trailing part of column $k$ to a multiple of the first coordinate in that block. It zeros every entry below the diagonal without changing earlier columns. Induction produces $H_n\cdots H_{1}A=R$ upper triangular.