Given a matrix group, how to determine whether its connected , dense etc.
As requested, here is an extended form of my comment.
For connectedness of a compact Lie group $G$ acting transitively on a space $X$ (say, a manifold), you have a fiber bundle $$ G_x\to G\to X, x\in X. $$ Hence, if $X$ is connected and $G_x$ is connected, then $G$ is also connected. This can be viewed as a special case of the long exact sequence of homotopy groups of a fibration (see e.g. Hatcher's "Algebraic Topology").
For density, one needs much more information.