Star-jacobian Matrix of Star with α Coefficient ★_{α}

**MLA Style: **Mohamed Moktar Chaffar, Hassen Ben Mohamed. "Star-jacobian Matrix of Star with α Coefficient ★α" International Journal of Mathematics Trends and Technology 67.2 (2021):85-102. **APA Style: **Mohamed Moktar Chaffar, Hassen Ben Mohamed(2021). Star-jacobian Matrix of Star with α Coefficient ★α. International Journal of Mathematics Trends and Technology, 85-102.

**Abstract**

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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**Keywords : **Coefficient star-matrix, algebra matrix, algebra linear equations matrix, Star-Jacobian matrix, Star.