Manifold structure on a set of matrices

Question

Let $Ω$ denote the $((n+1)×(n+1))$ matrix given by \begin{bmatrix} -1&0\\0&I_n \end{bmatrix} where $I_n$ is the $n×n$ identity matrix.

Let $X=\{A∈M_{n+1}(\mathbb{R}):A^tΩA=Ω\}$. How can I show that $X$ is a manifold? What will its dimension be? I am thinking of using the determinant map. It is easy to see that the determinant of each element of $X$ is $1$ or $-1$. What next? Any help will be appreciated.

Answer 1

$X$ is a group, so a neighbourhood of the identity matrix $I$ is mapped to a neighbourhood of any $A \in X$ by the map $B \mapsto AB$. So it suffices to prove that a neighbourhood of $I$ is a manifold. This can be done using the implicit function theorem.