Hadamard's Theorem

Let $\vert{\hbox{\sf A}}\vert$ be an $n\times n$ Determinant with Complex (or Real) elements $a_{ij}$, then $\vert{\hbox{\sf A}}\vert\not=0$ if

$$\vert a_{ii}\vert>\sum_{\scriptstyle j=1\atop\scriptstyle j\not=i}^n \vert a_{ij}\vert.$$

See also Hadamard's Inequality

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1110, 1979.