In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite field K of characteristic 3, satisfy all identities of the algebra M-2(K) of all 2 x 2 matrices over K? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebra M-2(K) over an infinite field K of positive odd characteristic, and to conjecture bases of the identities of M-2(K).122305316