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This question already has an answer here:

Let $A \in {M_n}$ and $\left| {{a_{ii}}} \right| > \sum\limits_{j \ne i} {\left| {{a_{ij}}} \right|} $.Why does $A$ is nonsingular?

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marked as duplicate by user147263, Claude Leibovici, Joel Reyes Noche, Algebraic Pavel, TravisJ Jun 2 '15 at 8:52

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  • $\begingroup$ You should write $j \ne i$, not $i \ne j$, since you're summing over $j$. $\endgroup$ – Keith Jun 2 '15 at 3:59
  • $\begingroup$ Dear @Kavir I see that, although you have already asked 5 question in this site and received answers in most of them, you have not mark a best answer in any of them. You can do it so by clicking on the checkmark next to the answer that you think is the one that helped you the most. Please read here for more detail. $\endgroup$ – Leo Sera Jul 2 '15 at 21:31
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Hint. Let $X = (x_j)$ be a nonzero column matrix. Prove that $AX \ne 0$ by considering the index $s$ for which $x_s$ has the largest absolute value.

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