The transition probability matrix of Markov Chain is

Find the limiting probabilities.

To find the limiting probabilities i.e $lim_{n→∞}⁡ {P^n}$

Let the limiting probabilities be π=[$π_1 π_2 π_3$ ]. Then we have πP=π such that $∑π_i=1$

∴[$π_1 π_2 π_3$ ]=[$π_1 π_2 π_3$ ]

$0.5π_1+0.3π_2+0.2π_3=π_1$ (1)

$0.4π_1+0.4π_2+0.3π_3=π_2$ (2)

$0.1π_1+0.3π_2+0.5π_3=π_3 $ (3)

We know$∑π_i=1$

$π_1+ π_2+ π_3=1$ (4)

i.e $ π_3=1-π_1-π_2$

Substituting the above value in equation (1) and (2)

$0.5π_1+0.3π_2+0.2(1-π_1-π_2)=π_1$

∴$-0.7π_1+0.1π_2=-0.2 $ (5)

$0.4π_1+0.4π_2+0.3(1-π_1-π_2)=π_2$

∴$0.1π_1-0.9π_2=-0.3(6)$

Multiply 9 with equation (5) and adding equation (5) & (6) we get,

$-6.2π_1=-2.1$

**∴$π_1$=0.3333

∴$π_2$=0.3703**

∴$π_3=1-π_1-π_2$

$π_3$=1-0.3333-0.3703

$π_3$=0.2964 ∴ $ π_1$=0.3333 $π_2$=0.3703 $ π_3$=0.2964