Abstract: We introduce a new computational technique for $n\times n$ matrices, over a $\Bbb{Z}_{2}$-graded ring $R=R_{0}\oplus R_{1}$ with $R_{0}\subseteq Z(R)$, leading us to a new concept of 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.
Keywords: $\Bbb{Z}_{2}$-graded ring, skew polynomial ring, determinant and adjoint
Classification (MSC2000): 16A38, 15A15; 15A33
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