Cayley-Hamilton theorem for 2 × 2 matrices over the Grassmann algebra Original Research Article

It is shown that the characteristic polynomial of matrices over a Lie nilpotent ring introduced recently by Szigeti is invariant with respect to the conjugation action of the general linear group. Explicit generators of the corresponding algebra of invariants in the case of 2 × 2 matrices over an algebra over a field of characteristic zero satisfying the identity [[x, y], z] = 0 are described. In this case the coefficients of the characteristic polynomial are expressed by traces of powers of the matrix, yielding a compact form of the Cayley-Hamilton equation of 2 × 2 matrices over the Grassmann algebra.