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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 &lt; 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.

On some properties and special identities in the second order matrix algebra over Grassmann algebras