Order of the sum of elements of the inverse of a matrix

Question

For each $T$, let $A_T$ be a $T\times T$ matrix of real numbers. let $e_T$ be the $T\times 1$ vector of ones. Assume that the sum of all entries of the matrix $A_T$ divided by $T^2$ is limited as $T$ goes to infinity. That is, $e_T'A_Te_T=O(T^2)$. Based on that assumption, can I say something about the limiting behavior of the sums $e_T'A_T^{-1}e_T$ or $e_T'A_T^{1/2}e_T$? For example, is $e_T'A_T^{-1}e_T=O(1/{T^2})$? or $e_T'A_T^{1/2}e_T=O(T)$? Are there any relations of this kind?