Further matrix algebra

M1 $\det(\mathbf{A} - 4\mathbf{I}) = -2(2 - 1) - 1(-2 - 0) = 0$ , so $4$ is an eigenvalue (1)
M1 Solves $(\mathbf{A} - 4\mathbf{I})\mathbf{x} = \mathbf{0}$ : $-2x + y = 0$ , $x - y + z = 0$ (1)
A1 Eigenvector $\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$ (or any non-zero multiple) (1)
M1 Normalises each eigenvector (divides by its magnitude) (1)
A1 Unit vectors $\tfrac{1}{\sqrt{3}}(1, -1, 1)$ , , (1)
A1 $\mathbf{P} = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ -\frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \end{pmatrix}$ (1)
A1 $\mathbf{D} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{pmatrix}$ , with columns of and in matching order (1)

Specification points