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

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


Does it have something to do with the Leibniz formula?

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

where  \sigma is a permutation.