Positive integer solution of the matrix equation X<sup>n</sup> = A for third-order matrices in the case of positive integers n
The problem of the positive integer solution of the equationXn = A for different-order matrices is important to solve a large range of problems related to the modeling of economic and social processes. The need to solve similar problems also arises in areas such as management theory, dynamic programming technique for solving some differential equations. In this connection, it is interesting to question the existence of positive and positive integer solutions of the nonlinear equations of the form Xn = A for different-order matrices in the case of the positive integer n. The purpose of this work is to explore the possibility of using analytical methods to obtain positive integer solutions of nonlinear matrix equations of the form Xn = A where A, X are the third-order matrices, n is the positive integer. Elements of the original matrix A are integer and positive numbers. The present study found that when the root of the nth degree of the third-order matrix will have zero diagonal elements and nonzero and positive off-diagonal elements, the root of the nth degree of the third-order matrix will have two zero diagonal elements and nonzero positive off-diagonal elements. It was shown that to solve the problem of finding positive integer solutions of the matrix equation for third-order matrices in the case of the positive integer n, the analytical techniques can be used. The article presents the formulas that allow one to find the roots of positive integer matrices for n = 3,...,5. However, the methodology described in the article can be adopted to find the natural roots of the third-order matrices for large n.