Hadamard's Inequality

Let ${\hbox{\sf A}}=a_{ii}$ be an arbitrary $n\times n$ nonsingular Matrix with Real elements and Determinant $\vert{\hbox{\sf A}}\vert$, then

$$\vert{\hbox{\sf A}}\vert^2\leq \prod_{i=1}^n\left({\,\sum_{k=1}^n {a_{ik}}^2}\right).$$

See also Hadamard's Theorem

References

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