The paper considers the anticommutative multiplication property for the matrix algebra M 2 ( G ) over the infinite Grassmann algebra G . We define as well some classes of *-symmetric and * skew-symmetric matrices in ( M 2 ( G ), * ) for * being the transpose or the symplectic involution. As a consequence we give some *-identities for M 2 ( G ) of degree 8. Examples of the application of finite-dimensional Grassmann algebras in quantum field theory are mentioned and some properties of the algebras G 4 and M 2 ( G 4 ) are given as well.
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