ON TRIANGULAR DECOMPOSITIONS TO EVALUATE THE DETERMINANT OF AN ARBITRARY SQUARE MATRIX A INCLUDING THE SOLUTION OF Ax=b AND ON THE RELATED COMPUTATIONAL RECURRENCE RELATIONS

Abstract

The invariant nature of the determinant of a matrix under elementary taonsformations affords an easy determining way of obtaining the determinant of a matrix by converting it into a trangular form the single LR decomposition does it still more easily, provided all the leading submatrices are non-singular In fact, in case of any singular or near-singular leading subtnatrix or submatrices LR decomposition fails or produces inaccurate results. In this context, presented in this paper are four types of trianguiar decompositions (differnt from LR typs) which usually succed in such singular on near-singular cases.Also presented here is a discustion of all possible triangular decompositions and their usefulness and uniqueness in presening a square matrix. A solution of a system of linear algebraic equations Ax=b through the easy inversion of the trtangular ones is shown. Also presented are the simple explint computational recurrence relations for easy antomatic computations of the solution vector x of Ax=b by all possible triangular decompositions. A typical numerical example has been worked out as an illustration.