eigenvalue column sum equals one linear algebra

Question

What does the following imply about the eigenvalues of an $n\times n$ matrix $A$:

The sum of the entries in each column equals $1$.

Thank you!

Answer 1

Hint: Prove that if the sum of every row is a constant $k$ then $k$ is an eigenvalue of $A$ with a corresponding eigenvector $(1,...,1)^{T}$.

Recall that $k$ is an eigenvalue of $A$ if and only if $k$ is an eigenvalue of $A^{T}$