Space of tangents of a matrix group G?

Question

Given a smooth path A(t) through the identity in any matrix group G, how would one prove that the smooth path through any g in G, is of the form gA(t)? It is clear that gA(t) is differentiable and hence, a smooth path, however, I would have to prove that such a form is unique. I'm not sure exactly how to do that. I tried letting the smooth path through g be B(t), and showing that it must be equal to gA(t), but I'm kind of stuck.

Answer 1

If $\alpha$ is a path with $\alpha(0) = g$, then $g^{-1}\alpha$ is a path through the identity. Now multiply that path with $g$.