Consider a square $N$ invertible matrix $M$ where $N=100...1000$. The question was to compute the numerical value of the inverse of the matrix $M$. The issue was, the entry of $M$ ranged at large scale separated orders, between $10^{-200}$ to $10^{200}$, so when terms multiply together $5.5\times 10^{-200}+1.5\times (10^{-200})^2$ were essentially zero or infinity with respect to each other. Thus, the usual methods such as the Gaussian elimination will not return an actual inverse matrix, but rather a matrix with large scale separated remainders, i.e.
$$\text{max entry}(\text{Gaussian Elimination Inverse}(M)M)=1.6\times 10^{150}$$
What's the best way to compute the matrix inverse at scale separated orders?