Eigenvalue of $AB$ where $A,B$ have prescribed row sums

Question

Let $A$ and $B$ be $n\times n$ matrices such that the sum of elements of each row of $A$ is $1$ and the sum of elements of each row of $B$ is $2$.

Prove that one eigenvalue of $AB$ is $2$.

My try is that one eigenvalue of $A$ is $1$ since the sum of each row is $1$ and similarly eigenvalue of $B$ is $2$ but I can't prove that eigenvalue of $AB$ is $2$.

Answer 1

Hint: Prove that if $A$ and $B$ have a common eigenvector $v$, with associated eigenvalues $\lambda_A$ and $\lambda_B$, then $v$ is also an eigenvector of $AB$ with associated eigenvalue $\lambda_A\lambda_B$.