On identities related to generalized fibonacci and lucas numbers by the help of matri̇ces

Leonardo Fibonacci is also known as Leonardo Pisano or Leonard of Pisa. He is recognized as the youngest and most successful mathematician of the European Middle Ages. His father arranged for Fibonacci to receive his early education in Bugia, Algeria, to follow the family trade and learn the art of calculation. Here Fibonacci was introduced to Indo-Arabic calculus techniques. He wrote the book Liber Abaci, explaining how to perform mathematical calculations in the Indo-Arabic system. Undoubtedly, the number sequence 1,1,2,3,5,5,8,13,21,... mentioned in Liber Abaci is what made Fibonacci famous. This sequence of numbers is known as the Fibonacci sequence. The elements of this sequence are called Fibonacci numbers. The sequence of numbers 2,1,3,4,4,7,11,18,... when the initial conditions are chosen differently is called the Lucas sequence. Many identities have been obtained as a result of the studies on these two sequences in the literature. Matrices, mathematical induction and Binet formulas were used in the proof of these identities. In the first section, some information about the content of the study is briefly given. In addition, some works in the literature on quadratic matrices are introduced and how these matrices are used to derive the identities for the generalized Fibonacci and Lucas numbers are mentioned. In the second section, first the definitions of generalized Fibonacci and Lucas numbers are given. Then, some new 2×2 matrices with generalized Fibonacci number and generalized Lucas number were obtained with the matrices obtained from here. Then some 2×2 matrices with terms {0,1,±k,t,2t,±kt,k^2+2t} were selected and the nth power of these matrices was calculated. With the help of these matrices, new identities similar to the identities given above were obtained. In the fourth section, 3×3 dimensional matrices are included. In this section, it is mentioned how 3×3 dimensional matrices taken from some sources are used to obtain identities. Then some 3×3 matrices are given and their nth power is calculated using diagonalization method. These matrices were used to obtain new identities. Some of these identities are given below. The fifth section contains conclusions and recommendations.