 Solution

The second Givens rotation leaves row three unchanged, so $\boxed{c_{31}=b_{31}=0}$. Its other relevant entry is $c_{21}=b_{11}\sin {\theta }_2+b_{21}\cos {\theta }_2$. For $r_2=\sqrt{b_{11}^2+b_{21}^2}>0$, takeThen $c_{21}=0$ and $c_{11}=r_2$. If both entries vanish, choose ${\theta }_2=0$. The first column now has zeros below its diagonal.