This exercise explains why \(\lambda=1\) is an eigenvalue of a stochastic matrix \(A\text{.}\) To conclude that \(\lambda=1\) is an eigenvalue, we need to know that \(A-I\) is not invertible.

  1. What is the product \(S(A-I)\text{?}\)

  2. What is the product \(S\evec_1\text{?}\)

  3. Explain why \(\evec_1\) is not contained in the column space \(\col(A-I)\text{.}\)

  4. Explain why we can conclude that \(A-I\) is not invertible and that \(\lambda=1\) is an eigenvalue of \(A\text{.}\)