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The set of all 2 × 2 real matrices does not form a group under matrix multiplication, as the singular matrices among them are not 'invertible' matrix theoretically. However, if some specific subsets from those singular matrices be chosen, they do form a group under matrix multiplication, where the group theoretic inverse of the matrix theoretically non-invertible matrices sometimes turn out to be their corresponding Moore-Penrose inverses. In the first part of this paper we investigate the whole family of 2×2 real singular matrices and linear algebraically pin down those specific subclasses. Now, 2 × 2 real matrices are nice geometric objects, as their realization in the guise of linear transformation from R² to itself can really be visualized on the Cartesian plane R² through pictorial presentations via the fundamental subspaces of those matrices. This natural interplay between linear algebra and its pictorial geometric presentation available in the present context then motivates us to see these different subclasses of matrices geometrically on R², an odyssey through the beautifully synchronized and harmonious geometric presentation of their locations, finally culminating in a well-cataloged library of all these 2×2 real singular matrices with respect to the above mentioned property. [ABSTRACT FROM AUTHOR]