Lemma 55.2.1. Let $A = (a_{ij})$ be a complex $n \times n$ matrix.

If $|a_{ii}| > \sum _{j \not= i} |a_{ij}|$ for each $i$, then $\det (A)$ is nonzero.

If there exists a real vector $m = (m_1, \ldots , m_ n)$ with $m_ i > 0$ such that $|a_{ii} m_ i| > \sum _{j \not= i} |a_{ij}m_ j|$ for each $i$, then $\det (A)$ is nonzero.

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