Based on the equality between the schein rank of matrix and the rank of matrix on Soft Algebra [0,1], this paper constructively demonstrates that the unique generalized inverse matrix exists in any non-zero matrix on [0,1].

By means of the rank of matrix, line outspreading, it gives some conditions in which a matrix can decompose to two Kronecker products of matrix.

In this note,we describe the equivalent propositions on the rank of matrix by determinants,equivalent of matrix,system of linear equations,linear space,linear mapping and so on.

Necessary and sufficient conditions for the Frobenius inequality of rank of matrix to be equality are dicussed in this paper,and the characterization of rank of a class of matrix is characterized.

This paper summarizes the applications of elementary transformation of matrix in solving the rank of a matrix or a set of vectors,calculating inverse matrix or system of linear equations,and solving the system of linear equations and the greatest common divisor of polynomials with examples,furthermore,it introduces the thought and application of generalized elementary transformation.

There are many proofs of this theorem: matrix order equals row order of matrix equals line order of matrix.