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In this paper a new block-structure method is presented for the solution ofthe well-known from gravity theory matrix system of equationsg{ij}g{jk}=delta{i}{k} (with respect to the unknown covariant components g{ij}and by known contravariant ones g{jk}) by transforming this matrix system intoa linear algebraic system of equations in the general N-dimensional case.Although powerful computer methods exist for the solution of this problem for agiven (fixed) dimension of the matrices g{ij} and especially for numericalelements of g{ij}, the structure of the obtained linear algebraic system in thegeneral N-dimensional case and for arbitrary elements of g{ij} (functions) hasnot been known.  The proposed new analytical block-structure method for the case ofsymmetrical matrices g{ij} and g{jk} (the standard case in gravity theory) isbased on the construction of a block-structure matrix, whose "elements" areagain matrices. The method allows to obtain the structure of this linear systemin the general N-dimensional case, after multiplication (to the left) with thetransponed matrix.  Some arguments are given why the proposed method may be applied, after somerefinement and generalization for the case of non-symmetrical matrices g{ij}and g{jk}, for finding the graviton modes in the Kaluza-Klein expansion intheories with extra dimensions.

Block-Structure Method for the Solution of the Matrix System of Equations g{ij}g{jk}=delta{i}{k} in the N-dimensional Case