Why are eigenvectors of Matrices orthogonal to the vector 1

Question

For transposes of Markov matrices we have eigenvalue $\lambda = 1$ with eigenvector $\vec{1}$. According to my professor, all eigenvectors with eigenvalue other than 1 of the Markov matrix must be orthogonal to $\vec{1}$. Why is this the case?

Answer 1

Sum of each column in a Markov matrix $M$ is $1$.

So ${\bf1}^TM={\bf1}^T$.

If $M{\bf v}=\lambda{\bf v}$ then $${\bf1}^T{\bf v}={\bf1}^TM{\bf v}={\bf1}^T\lambda{\bf v}\ .$$ If $\lambda\ne1$ then ${\bf1}^T{\bf v}=0$. that is, ${\bf1}\cdot{\bf v}=0$.