Ensuring that a symmetric matrix with nonnegative elements is positive semidefinite

It is not sufficient to have positive diagonal entries. To see this, consider the matrix $$ A=\pmatrix{1& 10\\10& 1}. $$ It has the negative eigenvalue $-9$ to the eigenvector $$ v=\pmatrix{1\\-1}, $$and is thus not positive semi-definite.

Note however that a diagonally dominant symmetric matrix is positive semi definite.

In your case it is sufficient to add the condition $x \ge \sum_{j\neq i} A_{ij}, i\neq j$ to ensure $A$ to be positive semi definite.