Let $\displaystyle A \in \textbf{C}^{m\times m}$ , and $\displaystyle a_{j}$ be its j-th column.

We prove $\displaystyle \left| det (A) \right| \leq \prod ^{m}_{j=1} \left\| a_{j} \right\|_{2} $

Does it have something to do with the Leibniz formula?

$\displaystyle |det(A) | = \sum _{\sigma \in S_{n}} Sgn(\sigma ) \prod ^{n}_{i=1} a_{i,\sigma (i)} $

where $\displaystyle \sigma $ is a permutation.