##### Exercise8

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 $$A\text{.}$$

in-context