Define a Householder transformation $\mathrm{H}$ and show that it is an orthogonal matrix. Briefly explain how these transformations can be used for QR factorisation of an $m \times n$ matrix.

Using Householder transformations, find a QR factorisation of

$$A=\left[\begin{array}{rrr} 2 & 5 & 4 \\ 2 & 5 & 1 \\ -2 & 1 & 5 \\ 2 & -1 & 16 \end{array}\right]$$

Using this factorisation, find the value of $\lambda$ for which

$$A x=\left[\begin{array}{c} 1+\lambda \\ 2 \\ 3 \\ 4 \end{array}\right]$$

has a unique solution $x \in \mathbb{R}^{3}$.