Identify one row or column of inverse of a matrix (sparse or full)

Question

Let $\textbf{A}$ be a large (sparse) matrix ($size > 10^6\times 10^6$). How to compute $j^{th}$ row or column of $\textbf{A}^{-1}$ at low computational time.

PS: I need to compute only one row of $\textbf{A}^{-1}$.

Answer 1

The $j$th column of $A^{-1}$ is the vector $v$ that solves the equation $Av = e_j$, where $e_j$ is the standard basis vector with a $1$ in the $j$th entry and $0$s elsewhere. (Indeed, multiplying by a matrix on the right is like multiplying by many non-interacting column vectors side by side.) Therefore you can apply any known algorithm for solving $Av=e_j$ for $v$.

As for rows, just take transposes of everything first and do the same thing.