let A and B be $\displaystyle \color{red}n \times n$ real (we don't need this assumption!) matrices such that sum of the entries in each row of A is 1 and sum of the entries in each row of B is 2.
then show that 2 is an eigenvalue of AB?
let $\displaystyle x=\begin{bmatrix}1 & 1 & . & . & . & 1 \end{bmatrix}^T.$ then $\displaystyle Ax=x$ and $\displaystyle Bx=2x.$ thus $\displaystyle (AB)x=A(2x)=2Ax=2x.$