There are non-strictly diagonally dominant matrices which are nonsingular .
However, I found no information or reference to the following question: are all non (strictly or not) diagonally dominant matrices also singular?
(The Gershgorin circle theorem does not provide a definite answer on any eigenvalue being zero in this case)
After doing a search I found that something similar (but not quite as general) was asked here before , 5 years ago, but received no answer. Maybe someone can answer it now.