 Hadamard determinant inequality

For a square matrix with row vectors $r_i$, the absolute determinant is at most the product of the Euclidean norms of its rows. If the rows are dependent the determinant is zero. Otherwise Gram-Schmidt process leaves the determinant unchanged and replaces each row by its component orthogonal to its predecessors. The resulting orthogonal rows have lengths no greater than the original rows, and their volume is their product. This proves the inequality; equality holds for mutually orthogonal rows.