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This paper proposes a new elementary operation approach (EOA) to multidimensional (-D) realization and linear fractional representation (LFR) modeling for multi-input and multi-output (MIMO) -D systems, as an extension of the new EOA proposed for the single-input and single-output (SISO) case by the authors recently. It is shown that, due to the substantial differences between the SISO and MIMO systems, the extension is not straightforward and further significant development is necessary. A matrix relation property among the associated matrices under the augmenting and admissible elementary operations is first revealed. Based on this matrix relation property, the realization problem for the MIMO -D case is formulated as an elementary operation problem of a certain -D polynomial matrix, which makes the extension possible. General constructive procedures are then established for the regular realizations based on the right and left matrix fraction descriptions (MFDs) of a given transfer matrix, respectively, such that one can easily implement this approach by a computer program in, e.g., MATLAB or Maple. Numerical and symbolic examples are provided to illustrate the main ideas and the effectiveness of the proposed approach.