The aim of the paper is demonstration of equality max σ ∈ S n σ ≠ id { 1 , … , n } { ∏ i = 1 n | 〈 x i , x σ ( i ) 〉 | } = max τ ∈ S n τ is a transposition { ∏ i = 1 n | 〈 x i , x τ ( i ) 〉 | } for each sequence of vectors x 1 , … , x n in a unitary space. The obtained equality has an important application in optimizing of certain generalization of the Hadamard's inequality proved by Xiaodong Zhang and Shangjun Yang and by the authors of the paper independently. Some illustrating examples are presented in the article. Also a problem connected with the mentioned equality is formulated.