## determinant of a matrix and product of column norms

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.