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The current thesis focused on Control Systems, and in particular through a survey, demonstrates the usefulness of algebra and its functionality with Diophantine Equations in the analysis, synthesis and design for a wide range of different systems, such as: linear, non-linear, time varying, continuous time, discrete time, multivariable. Additional, some methods of resolving Diophantine Equations of Matrices are analyzed for multivariable systems. Also, indicated the unnecessary use of processing resources for the zeros elements. At this point, an attempt is made with polynomials matrices to gain time from the execution of the calculation product, which ultimately leads to the Zero Elements (Z.E.) method and to the development of a corresponding algorithm where finally only the necessary (non zero) elements are used to reach the result A.B = C. Furthermore, for the product of any two matrices A.B, the Z.E. O(n3-nz) method is compared with Strassen algorithm O(n2.807) and Williams O(n2.373), where n is the dimension of square matrix and z the number of zero elements in matrix A, and it is shown that the number of zeros in some cases, such as diagonal matrices, renders the Z.E. method more efficient than the others. Finally, an Extended Zero Elements (E.Z.E.) method is expanded from the Zero Element method, where the extra gain of complexity is examined when zero elements is present not only in one, but in both matrices.