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Matrices over centrally \(\mathbb Z_2\)-graded rings. (English) Zbl 1016.15013
The authors introduce a new computational technique for \(n\times n\) matrices over a \(\mathbb{Z}_2\)-graded ring \(R=R_0\oplus R_1\) with \(R_0\subseteq Z(R)\), leading to a new concept of the determinant which can be used to derive an invariant Cayley-Hamilton identity. An explicit construction of the inverse matrix \(A^{-1}\) for any invertible \(n\times n\) matrix \(A\) over a Grassmann algebra \(E\) is also obtained.

15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A24 Matrix equations and identities
15A75 Exterior algebra, Grassmann algebras
15A09 Theory of matrix inversion and generalized inverses
15A15 Determinants, permanents, traces, other special matrix functions
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