Symmetric matrix, a>0 on diagonal and b<0 off diagonal, positive definite?

Question

this came up in a specific example that I'm working on, but I'm wondering under what conditions it's true in general.

Consider an nxn matrix, all of whose diagonal entries are a>0 and all of whose non-diagonal entries are b<0. Is the matrix positive (semi)definite?

In the specific example I'm interested in, n=10000, a=17.552 and b=-0.00175538. But I'm interested in the question in general too.

Thanks

Edit: The general 2x2 case is easy to work out by hand. There, the matrix is positive definite (respectively semidefinite) if and only if a+b>0 (respectively >= 0). But what about in general?

Answer 1

Not true. Just take $1$ along the diagonal and $-10$ off diagonal in a $2 \times 2$ matrix.

Answer 2

In general (see answer by Kavi Rama Murthy), this is not true, but if the diagonal values are large, and off diagonals are small, then the Gershgorin circles might give you enough info to prove that the matrix has only positive eigenvalues.