Lie Algebra of $GL_{+}(n,\mathbb C)$

Question

My question is if the Lie algebra of $GL_{+}(n,\mathbb C)$ = {$A \in \mathbb C^{n \times n} |\det(A) > 0$} is the same as of $GL(n,\mathbb C)$. Futhermore I have a problems understanding what is the underlying manifold of these two Lie groups (are they also the same?).

Answer 1

For complex numbers, $GL_n(\Bbb C)$ is already connected, so there is only one connected component.

What's an easy way to show that $GL(n,\mathbb C)$ is connected?