This study focuses on the derivation of closed-form expressions for the entries of the matrix powers Sn4 (x, y), where S4(x, y) is a tridiagonal symmetric Toeplitz matrix associated with Fibonacci numbers Fn and Fn+1. Specific cases of the ordered pair (x,y), including (Fs+1, Fs), (F?s, F?(s+1)), and (F?(s+1), F?s), are investigated to characterize when S4(x, y) becomes a Fibonacci matrix. Using these closed-form expressions, we derive and analyze key matrix properties, including trace, determinant, and row sums. These results not only offer explicit evaluations of fundamental matrix characteristics, but also contribute to the theoretical understanding of Fibonacci matrices. As an application, the derived Fibonacci matrices are employed as key matrices in a Affine Hill cipher algorithm, highlighting their applicability in symmetric key cryptographic systems.
Köken et al. (Wed,) studied this question.