  We consider the solution of systems of linear algebraic equations (SLAEs)
with an ill-conditioned or degenerate exact matrix and an approximate
right-hand side. An approach to solving such a problem is proposed and
justified, which makes it possible to improve the conditionality of the SLAE
matrix and, as a result, obtain an approximate solution that is stable to
perturbations of the right hand side with higher accuracy than using other
methods. The approach is implemented by an algorithm that uses so-called
minimal pseudoinverse matrices. The results of numerical experiments are
presented that confirm the theoretical provisions of the article.
