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In relation to the undecidability for 3x3 matrices and the comment of http://mathoverflow.net/users/1587/john-stillwell that

"The corresponding problem for 2×2 matrices is apparently still open":

There is a recent proof that the membership for non-singular 2×2 integer matrices is decidable (i.e. for 2x2 integer matrices with nonzero determinant). http://arxiv.org/abs/1604.02303

However in terms of uncedaibiltity the identity problem for 3x3 integer matrices is still open, while the general open problem about the identity matrix was proved to be undecidable for 4x4 matrices over integers , see

Paul C. Bell, Igor Potapov: On the Undecidability of the Identity Correspondence Problem and its Applications for Word and Matrix Semigroups. Int. J. Found. Comput. Sci. 21(6): 963-978 (2010) and arxiv.org/abs/0902.1975

solving the long standing open problem see Problem 10.3 in http://press.princeton.edu/math/blondel/solutions.html Unsolved Problems in Mathematical Systems and Control. Theory, Princeton Univ. Press, 2004.

It also follows that whether a matrix semigroup is a group is undecidable for 4x4 integer matrices.