The real orthogonal matrix $\Omega^{[p, q]} \in \mathbb{R}^{m \times m}$ with $1 \leqslant p<q \leqslant m$ is a Givens rotation with rotation angle $\theta$. Write down the form of $\Omega^{[p, q]}$.

Show that for any matrix $A \in \mathbb{R}^{m \times m}$ it is possible to choose $\theta$ such that the matrix $\Omega^{[p, q]} A$ satisfies $\left(\Omega^{[p, q]} A\right)_{q, j}=0$ for any $j$, where $1 \leqslant j \leqslant m$.

Let

$$A=\left[\begin{array}{ccc} 1 & 3 & 2 \\ 1 & 4 & 4 \\ \sqrt{2} & 7 / \sqrt{2} & 4 \sqrt{2} \end{array}\right]$$

By applying a sequence of Givens rotations of the form $\Omega^{[1,3]} \Omega^{[1,2]}$, chosen to reduce the elements in the first column below the main diagonal to zero, find a factorisation of the matrix $A \in \mathbb{R}^{3 \times 3}$ of the form $A=Q R$, where $Q \in \mathbb{R}^{3 \times 3}$ is an orthogonal matrix and $R \in \mathbb{R}^{3 \times 3}$ is an upper-triangular matrix for which the leading non-zero element in each row is positive.