On the Unassociated Matrices Number of the n Order and a Given Determinant

Abstract

The main object of the present work is to derive new relations between the number of n-order non-associated matrices and a determinant N, which can subsequently be put into use. In this study we mainly employ the Hermite triangular form of n-order full matrices and the determinant N. The following new results are obtained in the work: 1.formula for σ0(n, p1• ⋯ • pk) n-order non-associated primitive matrices with non-square determinants values N= p1• ⋯ • pk, where pi are primes;2.formula for σ0(n, pα) primitive non-associated n-order matrices N= pα, where p is a prime;3.the recurrent relations is established for σ0(n, N) by order of matrices considered;4.an upper estimate for the number of the considered n order matrices and the determinant is obtained close to the precise value of σ(n, N) in the case where the canonical expansion of N is not given;5.the relationship between σ(n, pα) as well as σ0(n, pα) and the Gaussian coefficients by combinatorics is established.