Let $\boldsymbol x=[1,0,\cdots,0]^T\in\mathbb R^{n×1}$, $ \boldsymbol A\in\mathbb R^{n×m}$ be invertible, and let $f= |\boldsymbol A^{-1}\boldsymbol x|_1$, where $|(·)|_1$ is the absolute value norm and $\boldsymbol A^{-1}$ is any generalized inverse of $\boldsymbol{A}$.
Is it now possible to find a constant $ c $ based on the properties of $ \boldsymbol{A }$ such that $f<c_1$?
I have been looking at (induced) matrix norms, where it seems I may be able to get the result based on the eigenvalues and conditioning number of $\boldsymbol{A}$, however I am not able to finalize an answer. Feel free to make any other assumptions that are necessary.