Matrix inverse quesiton

Question

It's known that set of invertible matrices is dense in $M_n(\mathbb{F})$ and that the function taking an invertible matrix to its inverse is continuous. Given this, shouldn't we be able to define an inverse for any matrix by $A$ by taking $A_n\to A$ and then defining $A^{-1}=lim A_n^{-1}$?

Answer 1

Under this definition, what would be the inverse of the zero matrix? If we take $A_n$ to be the diagonal matrix with entries $1/n$ along the diagonal, then these converge to the zero matrix. But $$ A_n^{-1} = \begin{pmatrix} n & 0 \\ 0 & n \end{pmatrix} $$ does not converge as $n \to \infty$.