In this paper, we give some notes of AA*=A*A= (detA)E which is a basic formula in studying matrix in proving solution of linearized equations, special matrix and so on through examples.

The special matrix refers to the matrix whose elements are special in the numerical value or in the property.

Finally, using another identity of 2Φ 0 , the numbers of some special matrices over F q are also obtained.

The existence of solution for the linear complementarity problem is related with special matrices.

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.