# positive definiteness under fixed sum symmetric matrices

My question follows from another of my question: symmetric matrices and positive definiteness. Suppose the matrix $$A$$ is symmetric with all entries positive and each row sum is a fixed constant ($$>1$$). Assume that each diagonal entry is greater than the off-diagonal entries of the corresponding row (and column), by atleast 1 (that is, the difference between the diagonal entry and each corresponding off-diagonal entry is atleast more than 1).
From these conditions, can we conclude that $$A$$ is positive definite.
Try: I tried to write $$A=sI-B$$, where $$I$$ is the identity matrix and $$s$$ is the largest diagonal entry of $$A$$ ($$s>1$$, from the given conditions). Then, I am not sure how to proceed. I am not sure if the result follows by gershgorin's theorem. Any hint will be really helpful.

For a counterexample, let $$A = \pmatrix { 10 & 8 & 6 & 2 \cr 8 & 10 & 2 & 6 \cr 6 & 2 & 10 & 8\cr 2 & 6 & 8 & 10 }$$ and note that $$\det(A)=-3120$$.