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The structures of matrix algebra and geometric algebra are completely compatible and in many ways complimentary, each having their own advantages and disadvantages. We present a detailed study of the hybrid 2 × 2 matrix geometric algebra M(2,IG) with elements in the 8 dimensional geometric algebra IG=IG3 of Euclidean space. The resulting hybrid structure, isomorphic to the geometric algebra IG4,1 of de Sitter space, combines the simplicity of 2× 2 matrices and the clear geometric interpretation of the elements of IG. It is well known that the geometric algebra IG(4,1) contains the 3-dimensional affine, projective, and conformal spaces of Möbius transformations, together with the 3-dimensional horosphere which has attracted the attention of computer scientists and engineers as well as mathematicians and physicists. In the last section, we describe a sophisticated computer software package, based on Wolfram's Mathematica, designed specifically to facilitate computations in the hybrid algebra.