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$$ A \text{ is a Matrix in } \mathbb{R}^{n\times n} \text{ such that:} \left | a_{i,i} \right | > \sum_{\substack {j=1 \\j\neq i}}^{n} \left | a_{i,j} \right | \quad \text{ for all } i \in \left \{ {1..n} \right \} $$ Prove that A is invertible.

(By the way I'm looking for a solution that does not involve eigenvalues and determinants because we haven't studied them yet, we only studied linear maps and transformation matrices)

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marked as duplicate by darij grinberg, user10354138, Theo Bendit, YuiTo Cheng, Community Jun 10 at 14:42

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