Certainly the identity component is contained in $GL_n^+$. To see this, note that the image of the continuous map $\operatorname{det}:GL_n \to \mathbb{R}$ has two connected components, namely the positive real numbers and the negative real numbers. Since the determinant of the identity is positive, the whole identity component must have positive determinant.
To show that you get all of $GL_n^+$, you can try to write down a path from an arbitrary matrix in here to the identity through other positive determinant matrices. To solve this, you usually use some sort of matrix factorization technique. For instance, you can try looking at QR factorization.