Chapman-Kolmogorov equations (Markov chain)

[Win_odd Dhamnekar]

Is the above answer correct?if wrong, where it is wrong? What is the correct answer?

 

[blamocur]

Several comments:
I don't see how you get from before "Hence" to the formula after.
Since $Q_{i,j}$ is a probability it must be in the $[0,1]$ range, but I don't see how this can be true for arbitrary $P_{i,j}$ -- e.g. when $P_{i,j}=\frac{2}{3}$