If we assume that the first column (that is column 1) of the inverse matrix can be written as a vector with unknown entries $$ \mathbf{A}_1^{-1}= \begin{bmatrix} a_{11}^{-1} \\ a_{21}^{-1} \\ \dots \\ a_{n1}^{-1} \\ \end{bmatrix}, $$ then we have a linear set of equations $$ \mathbf{LU}\begin{bmatrix} a_{11}^{-1} \\ a_{21}^{-1} \\ \dots \\ a_{n1}^{-1} \\ \end{bmatrix} =\begin{bmatrix} 1 \\ 0 \\ \dots \\ 0 \\ \end{bmatrix}. $$