First we introduce the basic concepts of Pythagorean triplets and classify them into four different categories. These operator matrices demonstrate many classical properties on dealing them in connection to algebraic structural properties. In this paper our target is to introduce operator matrices (Pre, Post, and Shift) which when operator on Pythagorean and Fermat In this paper our target is to introduce operator matrices (Pre, Post, and Shift) which when operator on Pythagorean. In the case Pythagorean matrices in which the column entries are the entries of triplets (Right triangle) of consecutive integer, the shift operator matrix preserves the order and nature of original triangle. The same logic for square matrices having column (triplet) entries of fermat triangle. In some cases important algebraic properties including the result of product of n^th order are also mentioned.