# Question

A square matrix is called a stochastic matrix if all of its elements satisfy 0= pi, j = 1 and, furthermore

for all i. Every stochastic matrix is the transition probability matrix for some Markov chain; however, not every stochastic matrix is a valid two- step transition probability matrix. Prove that a 2 × 2 stochastic matrix is a valid two- step transition probability matrix for a two- state Markov chain if and only if the sum of the diagonal elements is greater than or equal to 1.

for all i. Every stochastic matrix is the transition probability matrix for some Markov chain; however, not every stochastic matrix is a valid two- step transition probability matrix. Prove that a 2 × 2 stochastic matrix is a valid two- step transition probability matrix for a two- state Markov chain if and only if the sum of the diagonal elements is greater than or equal to 1.

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