##### Exercise3

Determine whether the following statements are true or false and provide a justification of your response.

1. Every stochastic matrix has a steady-state vector.

2. If $$A$$ is a stochastic matrix, then any Markov chain defined by $$A$$ converges to a steady-state vector.

3. If $$A$$ is a stochastic matrix, then $$\lambda=1$$ is an eigenvalue and all the other eigenvalues satisfy $$|\lambda| \lt 1\text{.}$$

4. A positive stochastic matrix has a unique steady-state vector.

5. If $$A$$ is an invertible stochastic matrix, then so is $$A^{-1}\text{.}$$

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