Can an off-diagonal-negative matrix have both: a unique negative eigenvalue and a positive row-sum?
Solution 1:
Consider the matrix $$M = \begin{bmatrix}2 & -1 \\ -1 & 0\end{bmatrix}.$$ Clearly, $M$ is square and symmetric, and all of the off-diagonal entries of $M$ are negative. The sum of the elements in the first row of $M$ is positive. Also, the eigenvalues of $M$ are $1+\sqrt{2} > 0$ and $1-\sqrt{2} < 0$, so $M$ has a unique negative eigenvalue of multiplicity $1$..