On algebraic structures defined by 2D and 3D matrices

In this thesis different matrix products and the algebric structures on 2-dimensional and 3-dimensional matrices whose elements are selected from a division ring are examined. This thesis consists of seven chapters. The first chapter contains the introduction and purpose of the thesis and in the second chapter, the literature research related to the subject of thesis is given. In the third chapter, the basic definitions and concepts related to matrice theory are given. In the fourth chapter of the study; Hadamard, Schur Kroneker and Cracovian Product from the well known products of matrices are examined and some properties of them are presented. In the fifth chapter: the vector spaces and the matrix ring of 3-dimensional matrices are examined and the examples for the finite dimensional vector spaces of 3-dimensional matrices and the bases of them are given. In the sixth section, 2- and 3-dimensional matrix rings on the field were examined according to the operations of the usual sum of matrices, a scalar multiple of one matrix and the Zaka matrix multiplication of two matrices and it is shown that this structure is a non associative commutative algebra. In the last section, the results and recommendations of the study are presented.