Exercise 3

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 $|\lambda| \lt 1$.

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

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