In system (2), define a matrix (F=left(f_{i j} ight)) to have a row for each transient state

In system (2), define a matrix \(F=\left(f_{i j}\right)\) to have a row for each transient state in the set \(D\) of all transient states and a column for each recurrent state in a given recurrence class \(C\); define \(T_{D C}\) to be the portion of the transition matrix of the chain whose rows are the rows of the transient states in \(D\) and whose columns are the states in \(C\); define \(T_{D D}\) to be the portion of the transition matrix corresponding to the transient states, and let \(\mathbf{1}\) be a square matrix consisting entirely of 1's, whose size equals the number of states in C. Argue that (2) in matrix form is

\[ F=T_{D D} \cdot F+T_{D C} \cdot \mathbf{1} \]

Execute this in Mathematica for the gambler's ruin problem of Example 2 for recurrence class \(\{8\}\), and make sure that you get the same equations for \(f_{i 8}\) as in that example.