The paper discusses the possibility of obtaining solutions with positive elements for a nonhomogeneous system of linear algebraic equations with a given quadratic matrix A and with an indefinite right-hand side under the assumption of positivity of their values. The investigation has been conducted on second and third order matrices. The two techniques of obtaining solutions have been proposed: the analytical technique and matrix technique.
The first technique has been developed for obtaining solutions of a system of inequalities. Its basis is comprised of elementary transformations, as a result of which some part of variables is removed from the system of inequalities, while simplifying analysis bound up with resolving the issue of existence of solutions for the system of inequalities. An algebraic criterion, which expresses necessary conditions of existence of solutions for the system of inequalities, has been proposed for this technique.
The second technique is based on principal properties of matrices: eigenvalues of matrices and eigenvectors. Sufficient conditions of existence of real solutions for the initial system of equations and sufficient conditions of impossibility of existence of such solutions have been obtained for this technique.
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