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On the Involutive Matrices of the kth Degree

In this study, the gradation of involutive matrices, whose definitions were given before, is conducted. The solutions of the equation x2 = 1 in real numbers are ±1. Meanwhile, in the solution of the equation xk = 1 in real numbers, there is always the number ±1 that is independent of the power of k 2 Z+. This feature, which is revealed by this equation in real numbers, is the subject of the research. In particular, the kind of situation in which the equation would display in the matrices is determined. Initially, the second-order square matrices are studied by obtaining some of their properties. Then, new cases arising from the known addition, subtraction, multiplication, scalar multiplication, and division operations of this set of second-order and quadratic involutive matrices are investigated. Some properties of third- and second-degree involutive matrices, which can be provided and exist, are emphasized. Comparisons are performed on the involutive matrices. Examples, theorems, and lemmas emerging
between these two types of matrices are given. New concepts are introduced to the literature by using linear matrix equations and multiplications by the matrices