Matrices – Prove Sum of Each Row Equals s, Then s is an Eigenvalue of A

Question

Let $A$ be an $n \times n$ matrix.

$i)$Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.

$ii)$Prove that if the sum of each column of $A$ equals $s$, then $s$ is an eigenvalue of $A$.

I think that being an eigenvalue of $A$ implies that $sv=Av$ for some vector $v$. Furthermore, I know that $[a_i] = s$ if we let $a_i$ denote the i-th row of $A$. However, I do not seem to be able to find a link between these two facts. Could anyone please help me out?

Answer 1

HINT: Calculate $Av$ when $v=(1,1,\ldots ,1,1)^t$, what can you say?