By using trace representation it is proved that the linear complexity of a shift register sequence equals to the rank of its rank matrix, and an enumeration formula for normal bases of a finite field over any of its subfield is derived.

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.

The way to determine the reflexive g-inverse of full rank matrix A was discussed.

Secondly,the randomly generating of full rank matrix and per-muta.

This paper discusses the way about how to get the reflexive general inverse matrix of a full rank matrix A, and generalize this way, gives the general way for not full rank matrix.

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.