In a similar way we can compute the unknow entries of the second column, $$ \mathbf{LU}\begin{bmatrix} a_{12}^{-1} \\ a_{22}^{-1} \\ \dots \\ a_{n2}^{-1} \\ \end{bmatrix}=\begin{bmatrix} 0 \\ 1 \\ \dots \\ 0 \\ \end{bmatrix}, $$ and continue till we have solved all \( n \) sets of linear equations.