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Steady state vector calculator

This calculator is for calculating the steady-state of the Markov chain stochastic matrix. A very detailed step by step solution is provided. This matrix describes the transitions of a Markov chain. This matric is also called as probability matrix, transition matrix, etc

You can see a sample solution below. Enter your data to get the solution for your question

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$$ \displaylines{ \mathbf{\color{Green}{Let's\;call\;All\;possible\;states\;as\;}} \begin{bmatrix} 1 & 2 & \end{bmatrix} \\ \\ \mathbf{\color{Green}{First\;we\;have\;to\;create\;Stochastic\;matrix}} \\ \\ Stochastic\;matrix\;=\;P= \begin{bmatrix} * & 1 & 2 & \\ \\ 1 & 0.5 & 0.5 & \\ \\ 2 & 0.8 & 0.2 & \end{bmatrix} \\ \\ \\ \\ \mathbf{\color{Green}{For\;steady\;state.\;We\;have\;to\;solve\;these\;equation}} \\ \\ X*P=X \\ \\ and\; x_{1}+x_{2} =1 \\ \\ Where\;X\;=\; \begin{bmatrix} x_{1} & x_{2} & \end{bmatrix} \\ \\ P= \begin{bmatrix} 0.5 & 0.5 & \\ \\ 0.8 & 0.2 & \end{bmatrix} \\ \\ \mathbf{\color{Green}{That\;is\;}} \\ \\ \Rightarrow \begin{bmatrix} x_{1} & x_{2} & \end{bmatrix} \begin{bmatrix} 0.5 & 0.5 & \\ \\ 0.8 & 0.2 & \end{bmatrix} = \begin{bmatrix} x_{1} & x_{2} & \end{bmatrix} \\ \\ and\; x_{1}+x_{2} =1 \\ \\ \mathbf{\color{Green}{Simplifying\;that\;will\;give}} \\ \\ x_{1}*(0.5)+x_{2}*(0.8)=x_{1} \\ \\ x_{1}*(0.5)+x_{2}*(0.2)=x_{2} \\ \\ x_{1}+x_{2} =1 \\ \\ \mathbf{\color{Green}{Simplifying\;again\;will\;give}} \\ \\ x_{1}*(-0.5)+x_{2}*(0.8)=0 \\ \\ x_{1}*(0.5)+x_{2}*(-0.8)=0 \\ \\ x_{1}+x_{2} =1 \\ \\ \mathbf{\color{Green}{Solving\;those\;will\;give\;below\;result...}} \\ \\ \begin{bmatrix} x_{1} & x_{2} & \end{bmatrix} = \begin{bmatrix} 0.615385 & 0.384615 & \end{bmatrix} } $$