Constructing a minimal basis of invariants for differential algebra $(2\times2)$ matrix

A basis of invariants is constructed for a set of second-order matrices consisting of the original matrix and its derivatives. It is shown that the presence of a derivative imposes connections on the elements of this set and reduces the number of elements of the basis, compared with the purely algebraic case. Formulas for calculating algebraic invariants of such a set are proved. A generalization of Fricke's formulas is formulated in terms of traces of the product of matrices of this set.