Hadamard’s inequality

Let $A=(a_{ij})$ with $1\leq i,j\leq n\in\mathbb{N}$ be a square matrix with complex coefficients. Then the following inequality holds:

|\det(A)|\leq\prod_{i=1}^{n}\left(\sum_{j=1}^{n}|a_{ij}|^{2}\right)^{\frac{1}{% 2}}.

Moreover, if $A$ is Hermitian and positive semidefinite, the following inequality holds:

\det(A)\leq\prod_{i=1}^{n}a_{ii},

with equality if and only if $A$ is a diagonal matrix.