We consider the graph $G$ represented in Figure 2. We recall that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the vertices to which it is connected. We assume that initially, the point is on vertex 1, so that $P ^ { ( 0 ) } = ( 1,0,0,0,0,0,0,0 )$. We denote $S _ { 1 } = \{ 1,3,6,8 \}$ and $S _ { 2 } = \{ 2,4,5,7 \}$.
Give the transition matrix $T$ of this graph and calculate
$$( 1,1,1,1,1,1,1,1 ) T .$$