Let $A$ be an $n\times n$ real matrix such that:
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$a_{ii}=0$ for all $i$, $1>a_{ij}\geq0$ for all $i,j$.
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$\sum_{j=1}^n a_{ij}\in (0,1]$ for all $i$.
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There is at least a row $i_0$ such that $\sum_{j=1}^n a_{i_0j}<1$.
Can we conclude that the maximum eigenvalue of $A$ is less than 1?
By Perron-Frobenius Theorem the maximum eigenvalue is greater or equal than $0$. Note also that, without part 3, one could take a stochastic matrix satisfying 1 and 2, which has always $1$ as an eigenvalue.
EDIT: I added the hypothesis $a_{ij}<1$ for all $i,j$. In this case, the example below does not work.
Best Answer
No. Consider for instance
$$\left[\begin{array}{ccc}0 & 1 & 0\\1 & 0 & 0\\1/2 & 0 & 0\end{array}\right].$$