Let the space of all matrices over $\mathbb R$ of size $ n^2 $ , with the natural metric of $\mathbb R^{n^2 }$.
Prove that there exist a neighborhood of the identity matrix, such that all the matrices in that ball have a square root, i.e $$ \exists \varepsilon > 0\,:\forall X \in B\left( {I,\varepsilon } \right)\,\,\exists \,Y:\,Y^2 = X . $$
Hint: Consider the function $ F(X) = X^2 $ and use the Inverse Function Theorem
Help with this problem !! )=