Several upper bound analysis for the maximum determinant of a matrix formed by 1 to <i>n</i> ²

Abstract This paper discusses the upper bound analysis for the determinant of a matrix constructed using the numbers 1 to n 2 . We propose four different methods to approximate the upper bound, including using the AM-GM inequality, Hadamard’s inequality, maximizing diagonal elements, and the Perron-Frobenius theorem. Our results provide important insights into the structure and properties of matrices that maximize their determinants.