Markov chain: join states in Transition Matrix

Question

I need to merge two states in the Transition Matrix:

For example: I have the matrix below

      A       B       C      D      E      F 
  A  0.5     0.4      0      0      0.1    0

  B  0.5     0.1      0.2    0.1    0.1    0

  C  0       0.1      0.9    0      0      0

  D  0       0        0      0.7    0.3    0

  E  0       0.2      0      0.7    0      0.1

  F  0       0        0      0.5    0      0.5

And I want to join the states D and E:

          A      B        C    (D+E)    F 
      A  0.5     0.4      0      ?      0

      B  0.5     0.1      0.2    ?      0

      C  0       0.1      0.9    ?      0

   D+E   ?       ?        ?      ?      ?

      F  0       0        0      ?      0.5

What are the formulas to obtain the row and column (D+E)?

Using the constraint: "the sum over column must be equal to 1" is simple to calculate the elements:

(A,(D+E))=0.2

(B,(D+E))=0.2

(C,(D+E))=0.1

(F,(D+E))=0.5

How can I calculate the elements of row ((D+E),i)?

Answer 1

The question can not be answered in general, i.e. there are no "formulas" to compute it. The reason being is that \begin{align*} A(D+E,i) & = P(i | D \cup E ) \\ & = \frac{P(i | D) P(D) + P(i | E) P(E)}{P(D)+P(E)}\\ & = \frac{P(D)}{P(D)+P(E)} A(D,i) + \frac{P(E)}{P(D)+P(E)} A(E,i) \end{align*} and the right hand side depends essentially on $P(D)$ and $P(E).$

For your example, the question can be answered easily, e.g. using Markov chain graphs, but this is rather a "homework" level.