A matrix with all diagonal entries between 0 and 1, all off-diagonal entries between -1 and 0 and each row summing to 0 has always $|det|\leq 1$?

Question

Let $A$ be an $n\times n$ matrix with $|a_{ij}|\leq 1 \forall i,j$. Suppose that $a_{ii}\geq0 \forall i$ and that $a_{ij}\leq0 \forall i,j$ such that $i\neq j$. Moreover, suppose that that $\sum_j a_{ij}=0$ $\forall i$. Does this imply that $|det(A)| \leq 1$?

Answer 1

If the rows all sum to $0$, then the all-ones vector is in the null space of $A$, so $A$ is singular and $\det(A) = 0$.