The aim of this article is to characterize the 2 x 2 matrices "X" satisfying X[superscript 2] = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers. The recommendations regarding the teaching of the identities given in this article can be presented in two cases. The first is related to the pedagogical aspect. The identities require the use of mathematical knowledge learned in different courses. By this way, the authors consider to affect students' motivation in a positive direction. The second recommendation is related to the qualification of these identities. Their identities are different from other known identities and their method exploits the properties of matrix algebra to find more general identities.