We discuss some fundamental properties of inverse semigroups of matrices, and prove that the idempotents of such a semigroup constitute a subsemilattice of a finite Boolean lattice,and that the inverse semigroups of some matrices with the same rank are groups. At last, we determine completely the construction of the inverse semigroups of some 2 x 2 matrices: such a semigroup is isomorphic to a linear group of dimension 2 or a null-adjoined group, or is afinite semilattice of Abelian linear groups of finite dimension, or satisfies some other properties.The necessary and sufficient conditions are given that the sets consisting of some 2 × 2 matrices become inverse semigroups.
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