The linear system

$$\left[\begin{array}{lll} \alpha & 2 & 1 \\ 1 & \alpha & 2 \\ 2 & 1 & \alpha \end{array}\right] \mathbf{x}=\mathbf{b}$$

where real $\alpha \neq 0$ and $\mathbf{b} \in \mathbb{R}^{3}$ are given, is solved by the iterative procedure

$$\mathbf{x}^{(k+1)}=-\frac{1}{\alpha}\left[\begin{array}{lll} 0 & 2 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 0 \end{array}\right] \mathbf{x}^{(k)}+\frac{1}{\alpha} \mathbf{b}, \quad k \geqslant 0$$

Determine the conditions on $\alpha$ that guarantee convergence.