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 \(A^{-1}\text{.}\)