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Consider a quantum mechanical system with three states. At each step a particular particle transitions from one state to a different state, so that the system obeys the difference equation xₖ₊₁ = Axₖ. Here xₖ = (X₁,X₂,X₃), where xi is the probability that the particle is in state i at step k.
Empirical data show that if the particle is in State 1, then it is 6 times more likely to go to State 2 at the next step than to State 3. If it is in State 2, then it is 5 times more likely to go to State 3 at the next step than to State 1. If it is in State 3 then it is equally likely to go to State 1 or State 2 at the next step.
Let A = (aᵢⱼ). Find a₃₁, a₃₂, and a₃₃

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