For a project, i was trying to check convergence of the iterative updates $$\begin{bmatrix} s_{1}\\ s_{2} \\ s_{3} \end{bmatrix}:=\begin{bmatrix} \frac{1}{s_{1} + s_{2} + 2s_{3}} + \frac{2}{2s_{1} + s_{2} + s_{3}} & \frac{1}{s_{1} + s_{2} + 2s_{3}} & 0 \\ \frac{1}{s_{1} + s_{2} + 2s_{3}} & \frac{1}{s_{1} + s_{2} + 2s_{3}} + \frac{1}{2s_{1} + s_{2} + s_{3}} & \frac{1}{2s_{1} + s_{2} + s_{3}} \\ 0 & \frac{1}{2s_{1} + s_{2} + s_{3}} & \frac{2}{s_{1} + s_{2} + 2s_{3}} + \frac{1}{2s_{1} + s_{2} + s_{3}} \end{bmatrix} \begin{bmatrix} s_{1}\\ s_{2} \\ s_{3} \end{bmatrix}$$
For any random initialization of $s_{1}, s_{2}, s_{3}$, all three seem to converge to 1. Is there any way I can find this convergence value (=$1$) or know when (for what properties of the $n$ x $n$ matrix) would $s_{1}, s_{2}, s_{3}$ all converge to the same number?
In general, is it possible to figure out all the fixed points that a system $Ax = b$ like above will converge to?