When an analysis of a concept is given like "A bachelor is an unmarried male." How is it decided if the analysis is correct and complete? Is there any way we can 'check' an analysis?
There is no consensus in regarding a normative framework for differentiating between 'complete' and 'incomplete' analysis of concepts. Some philosophers are highly skeptical of analysis and the linguistic turn entirely. In practice, philosophers largely rely on their intuitions or pragmatic factors on when to conclude an analysis, much in the same way a functional analysis might be conducted under Cummins' method of functional analysis.
Not only is there no universal framework or protocol for 'complete conceptual analysis', it's arguable that philosophy is essentially one, long continual analysis of concepts, while some philosophers argue against such a definition. Quine famously attacked the whole notion of analyticity in his Two Dogmas. In popular culture, philosophical discussion is often portrayed as an eternal rehashing of essentially contested concepts of which no genuine progress is ever made.
Furthermore, analysis is taken to be useful if it 'informative', that is produces some sort of new information. And yet, if an analysis is to be held 'correct', in some sense it needs to be complete, that is, all of the relationships it entails should be addressed. That the twin goals of informativeness and correctness are driving in different directions itself seems to be a problem. G.E. Moore considered such a matter to be a philosophical problem in its own right: the paradox of analysis. From WP:
A conceptual analysis is something like the definition of a word. However, unlike a standard dictionary definition (which may list examples or talk about related terms as well), a completely correct analysis of a concept in terms of others seems like it should have exactly the same meaning as the original concept. Thus, in order to be correct, the analysis should be able to be used in any context where the original concept is used, without changing the meaning of the discussion in context. Conceptual analyses of this sort are a major goal of analytic philosophy.
So, there is no one-size-fits-all approach to conceptual analysis. In fact, conceptual analysis, which is a mainstay of analytic philosophy, should provoke the idea that analytic philosophy is not the only style of philosophy. Many philosophers see, including analytic philosophers themselves, the incessant hair splitting of analysis to be a from of pedantry, one that has lost view of what is important in philosophy, such as the pursuit of eudaimonia.