Solution
= 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 $r_2=\sqrt{b_{11}^2+b_{21}^2}>0$, take
$$
\boxed{\cos\theta_2=\frac{b_{11}}{r_2},\qquad\sin\theta_2=-\frac{b_{21}}{r_2}.}
$$
Then $c_{21}=0$ and $c_{11}=r_2$. If both entries vanish, choose $\theta_2=0$. The first column now has zeros below its diagonal.