Inverse powers and 2-adic properties of binary 3 x 3 matrices associated with the Tribonacci sequence

Abstract

In this paper, we study binary 3 x 3 matrices with determinant one that generate the Tribonacci recurrence. Starting from a specific generating matrix, we obtain explicit expressions for its powers and inverse powers in terms of the Tribonacci sequence T_n and two associated sequences B_n and R_n. This matrix framework yields identities involving traces, determinants, and products, including addition formulas and Cassini-type relations linking T_n, B_n, and Rn. Moreover, we derive new identities from the matrix representation of inverse powers and establish structural relations between these sequences. As an application, we investigate arithmetic properties of Tribonacci numbers and determine the exact 2-adic valuation of T_2^k for sufficiently large k. Finally, we provide a classification of 3 x 3 matrices associated with the Tribonacci recurrence via similarity over Q.