The purpose of this article is to study determinants of matrices which are known as generalized Pascal triangles (see ). We present a factorization by expressing such a matrix as a product of a unipotent lower triangular matrix, a Töeplitz matrix and a unipotent upper triangular matrix. The determinant of a generalized Pascal matrix equals thus the… (More)
The purpose of this article is to prove several evaluations of determinants of matrices, the entries of which are given by the recurrence a i,j = a i−1,j−1 +a i−1,j , i, j ≥ 2, with various choices for the first row a 1,j and first column a i,1. 1. Introduction. Determinants have played a significant part in various areas in mathematics. For instance, they… (More)
The purpose of this article is to obtain some new infinite families of Toeplitz matrices , 7-matrices and generalized Pascal triangles whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We also present a new proof for Theorem 3.1 in [R. Bacher. Determinants of matrices related to the Pascal triangle. In memory of… (More)
Let φ = (φ i) i≥1 and ψ = (ψ i) i≥1 be two arbitrary sequences with φ 1 = ψ 1. Let A φ,ψ (n) denote the matrix of order n with entries a i,j , 1 ≤ i, j ≤ n, where a 1,j = φ j and a i,1 = ψ i for 1 ≤ i ≤ n, and where a i,j = a i−1,j−1 + a i−1,j for 2 ≤ i, j ≤ n. It is of interest to evaluate the determinant of A φ,ψ (n), where one of the sequences φ or ψ is… (More)
In this paper, we first find the set of orders of all elements in some special linear groups over the binary field. Then, we will prove the characterizability of the special linear group PSL(13, 2) using only the set of its element orders.
The spectrum ω(G) of a finite group G is the set of element orders of G. If Ω is a non-empty subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups G with ω(G) = Ω and put h(G) = h(ω(G)). We say that G is recognizable (by spectrum ω(G)) if h(G) = 1. The group G is almost recognizable (resp. nonrecognizable)… (More)