ESTIMATES OF DETERMINANTS AND OF THE ELEMENTS OF THE INVERSE MATRIX UNDER CONDITIONS OF OSTROVSKII THEOREM

In 1951 A. M. Ostrovskii published the remarkable theorem being the far generalization of the known J. Hadamard theorem concerning the matrix with the diagonal predom-ination [3]. The mentioned Ostrovskii result is the following. Let A = (aij) be the square matrix of the order n. Let for a certain value of the parameter α, 0 < α < 1, the following conditions be satisfied:∣∣aii∣∣> p1−αi qαi, i = 1,2,...,n, (1) where pi = j =i ∣∣aij ∣∣, qi =∑ j =i ∣∣aji∣∣, i = 1,2,...,n. (2) Then the matrix A is regular. If we put formally in (1) α = 0 or α = 1, we come to Hadamard theorem for matrices with predomination with respect to the row in the first and with respect to the column in the second case (see [2]). Thus, introducing the parameter α Ostrovskii succeeded in the happy connection of different type conditions and—that is maybe, more important—he opened the way of using Hölder inequality in investigations of tests of the matrix regularity. (Remember that the matrix is called regular if it has an inverse matrix. Denotion of the sum in (2) means that the summation is taken over all indexes j = 1,2,..., except j = i.) In the following presentation magnitudes σi =