The famous inequality in matrix theory of J. Hadamard has different proofs and extensions [1, 6]. Here given proof is by the method of common fixed-point of mappings monotonie with respect to a functional, which can be applied to many, including all main inequalities [3]. Condition for equality: rows of a matrix are orthogonal or at least one of them is zero, is replaced by proportionality (appearing in numerous other inequalities) between rows of a matrix and corresponding rows of cofactors.