Want to show any two matrices in $GL_n(\mathbf{R})^+$, i.e. $n\times n$ matrices with positive determinant can be connected by a path. Now It being a manifold path connectedness and connectedness are equivalent.
Using Gram-Schmidt we can show that $GL_n(\mathbf{R})^+$ deformation-retracts to $SO(n)$. Now, $SO(2)\cong S^1$, so connected. And $SO(n)/SO(n-1)\cong S^{n-1}$, so using induction I get $SO(n)$ is path connected. Hence $GL_n(\mathbf{R})^+$ is path connected. But it's all abstract, can anyone help with an explicit formula for such a path?