Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 40A / b / iii / Solution

For $k=1$ the claim is just $A=Q_0{R}_0$. Ifthen $A_k={\tilde{Q}}_k^{T}A{\tilde{Q}}_k=Q_k{R}_k$, and henceInduction proves $A^k={\tilde{Q}}_k{\tilde{R}}_k$ for all $k\ge 1$.