Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 2A / i / Solution

Left multiplication by the given Givens rotation mixes rows two and three, so $b_{31}=a_{21}\sin {\theta }_1+a_{31}\cos {\theta }_1$. If $r_1=\sqrt{a_{21}^2+a_{31}^2}>0$, chooseEquivalently ${\theta }_1=atan2(-a_{31},a_{21})$ modulo $2\pi$. Substitution gives $b_{31}=0$ and $b_{21}=r_1$. If $a_{21}=a_{31}=0$, take ${\theta }_1=0$; the desired zero is already present. This also covers singular matrices without division by zero.