In this paper, we define fuzzy matrices (FM), intuitionistic fuzzy matrices (IFM) and a new operation * defined on the set I-n = {1, 2, ... , n}. The system (I-n, *) is an abelian group. This group is pivotal in our paper. Circulant intuitionistic fuzzy matrices is defined also in this paper via the group (I-n, *) as a new way for defining circulant matrices. However, we study and investigate several properties of circulant intuitionistic fuzzy matrices. The first row (column) of a circulant matrix plays an important role in our studying. From any intuitionistic fuzzy matrix R (not necessarily circulant) we can construct two circulant intuitionistic fuzzy matrices R-L and R-U such that R-L <= R <= R-U. We also show that some compositions of circulant intuitionistic fuzzy matrices are also circulant. Moreover, we proved that RS - SR for two circulant intuitionistic fuzzy matrices R and S. However, we give some examples to clarify our results.