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Given a matrix $A=(a_{nm})^N_{n,m=1}$ with $$ \min\limits_n|a_{nn}| > (N-1) * \max\limits_{m \neq n}|a_{nm}| $$ then $A$ is regular and $$ \lVert A^{-1} \lVert_1 \leq \left ( \min\limits_n|a_{nn}| - (N-1) * \max\limits_{m \neq n}|a_{nm}| \right )^{-1} $$ holds.

I already showed that A is regular by using, that $A$ is strictly diagonally dominant. Now I want to show that the estimate holds.

Edit: So I found the following lemma in my lecture notes:
If $A$ is strictly diagonally dominant, then $A$ is regular and $$ \lVert A^{-1} \lVert_\infty \leq \max\limits_{1 \leq n \leq N} \left ( |a_{nn}| - \sum_{m \neq n} |a_{nm}| \right )^{-1} $$ My approach now is using the first equation to show, that $A^T$ is again strictly diagonally dominant. The second equation can then be easily achieved by some transformations of the third one.

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