The Representations of the Fibonacci and Lucas Matrices

In this study, a matrix R-L is defined by the properties associated with the Pascal matrix, and two closed- form expressions of the matrix function f (R-L) = R-L(n) are determined by methods in matrix theory. These expressions satisfy a connection between the integer sequences of the first-second kinds and the Pascal matrices. The matrix R-L(n) is the Fibonacci Lucas matrix, whose entries are the Fibonacci and Lucas numbers. Also, the representations of the Lucas matrix are derived by the matrix function f (R-L - 5I), and various forms of the matrix (R-L - 5I)(n) in terms of a binomial coefficient are studied by methods in number theory. These representations give varied ways to obtain the new Fibonacci- and Lucas- type identities via several properties of the matrices R-L(n) and (R-L - 5I)(n).