It's well known that $SL_2(\widehat{\mathbb{Z}})$ contains $SL_2(\mathbb{Z})$ as a dense and finitely generated subgroup. However, $GL_2(\mathbb{Z})$ is not dense in $GL_2(\widehat{\mathbb{Z}})$, since $GL_2(\mathbb{Z})$ is contained in the closed subgroup of matrices with determinant $\pm 1$, which is very far from the entirety of $GL_2(\widehat{\mathbb{Z}})$ (whose determinant map surjects onto $\widehat{\mathbb{Z}}^\times$)

Is there a finitely generated dense subgroup of $GL_2(\widehat{\mathbb{Z}})$?