Description

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A simple and systematic procedure for solving any k x k system of linear equations is developed in this paper. The determinant of the equation matrix is first found using Jibunoh's method. Then the matrix is inverted by applying the defined backward vector substitutions (bvs). The reciprocal of the positive value of the determinant, if the matrix is real, is taken as a factor of the inverse matrix. The complex matrix is similarly inverted to obtain what is defined as either the Analytical or Empirical inverse. The entries of any inverse matrix (real or complex) are mainly integers, without the scalar-factor multiplying the matrix. This makes the inverse matrix exact and more accurate than decimal representations obtained by computer evaluations. For any system of equations, therefore, three quantities are obtained simultaneously, namely, the determinant of the equation matrix, the inverse of the matrix and the solution of the system. The production of these quantities simultaneously, is new in the literature. By these procedures, any linear systems of equations of dimensions k can be solved easily and accurately, as k tends to infinity.