Abstract: Using Gauss elimination method with part main elements is generally not got all principal sub determinants of matrix.This article discussed around gradual reduction with each step-by-step subdivision,final triangular decomposition was performed on matrix A0 by row permutation after a number of row replacement,and each pre order principal sub determinant of A0 was found out orderly.Main purpose of the article is to achieve improvement for usually triangle reduction method by row permutation,with a recursive method for algebraic representation,to bind matrix product ope-ration,and it inductively proves that final reduction result is in accordance with the realization of the L-U triangular decomposition to matrix.And at the same time with the process of gradual reduction,we got all pre order principal sub determinants of original matrix A₀.

Key words: Row permutation,Gradual reduction,Triangular decomposition,Pre order principal sub determinant,Matrix operation