I am interesting in estimating the norm of the inverse of a real square matrix $A\in \mathbb R^{n\times n}$, which is defined by $$\|A^{-1}\|=\sup_{u\in \mathbb R^n}\frac{\|A^{-1}u\|}{\|u\|}$$
My idea is to search for the vectors $v_i\in \mathbb R^n$ such that $$Av_i=\vec e_i,\qquad i=1,2,...n$$ and then estimating the norm as $$S(A)=\sup_i \|v_i\|$$
Is it true that there exists a constant $c$ depending only on the dimension of the matrix such that $$\|A^{-1}\|\le cS(A)?$$