Volume 67 | Issue 2 | Year 2021 | Article Id. IJMTT-V67I2P513 | DOI : https://doi.org/10.14445/22315373/IJMTT-V67I2P513

In this paper, we introduce a new data type of Star-matrices and we define a simple basis Star-Jacobian vector that enables the representation of Star-Jacobian matrices (directly, indirectly) composed of the solution of a Star-System with α coefficient. The resolution of Star-Systems is laid in a basis of the known Gauss’ method (method of exception of unknown values) for solution of system of linear equations. When we solve a linear Star-System, we will place the 5 Star-Vectors that are the solution (linearly independent) in the columns of a matrix. The so-called fundamental matrix of the Star-System (5x5). Thereafter, we start with two examples with detailed solutions are presented. This can, in particular, be exploited to obtain arithmetic properties for classes of Star-Matrices. According to a number of different studies, we also note that there is a constant coefficient matrix Cα ★ if we multiply that matrix by M★+ (Star-Matrix directly). Then you will get the matrix M★- (Star-Matix indirectly). Cα ★ is an orthogonal matrix (t Cα ★ = (Cα ★ ) -1 ). On the other hand, we study the relationship between two Star-Jacobian matrices of Star with α coefficient, a relationship refers to the correspondence between two Star-Matrices. The results of calculations show that the products between two Star- Jacobian Matices of two Star-System (★α1, ★α2) with α1 and α2 coefficient (directly-indirectly) or (directly-directly) or (indirectly-indirectly) are diagonalizable.

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Mohamed Moktar Chaffar, Hassen Ben Mohamed, "Star-jacobian Matrix of Star with α Coefficient ★α," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 67, no. 2, pp. 85-102, 2021. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V67I2P513