On Some Identities Related to Generalized Fibonacci and Lucas Numbers

In this paper, we define some new matrices similar to the classical matrices introduced by Gould in [9]. We calculate the nth powers of the new matrices by diagonalizing them with the help of eigenvalues and eigenvectors. Thus, by making use of Binomial expansions, we obtain new identities containing generalized Fibonacci and Lucas numbers. These new results inform us about the relationships between matrix algebra and sequence theory, especially in the context of generalized Fibonacci and Lucas sequences.