Let G be a finite group. We define in this paper what is called G-compatible intuitionistic fuzzy matrices and we prove some of their fundamental properties. Of course, these matrices are square (since G is finite). However, the first row of our matrices play an important role in this study. The set of all G-compatible intuitionistic fuzzy matrices is a commutative semiring with respect to the operations ∨ and ∘, respectively. Also, we study the G-Min-compatible intuitionistic fuzzy matrices and prove some of their properties. We have also provide some examples to clarify our notions and results.