LU Decomposition, the inverse

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.

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