1. Elementary Operations of Matrix
Elementary operations of matrix are an important tools in linear algebra.
The next three operations are called as No.1st,2ndand3rd elementary transforms of matrix.

2. Function How to transform a matrix into an echelon one .
Elementary operations of matrix about row and column are
all called elementary operations
Function
How to transform a matrix into an echelon one .

4. Equivalent Matrix normal form of A
If matrix A can be transformed into B by some
elementary operations, we say A and B are equivalent,
and we marked as A B.
Equivalent Matrices have the following characters：
normal form of A
Theory: any matrix
has its own
normal form.

5. Corollary:matrices A and B are equivalent if
As last e.g.：
Corollary:matrices A and B are equivalent if
and only if they have the same normal forms.

6. Rank of Matrix 2.The highest order of nonzero subdeterminants
of A is called the rank of A and denoted by r(A).
Obviously:r(O)=0.As long as A is not 0, r(A)>0. And :

7. e.g.: Compute the rank of Matrix A.
We can figure out the rank of the matrix by elementary operations

8. The Rank:
Theory: Elementary operations do not change the resulting scalar of the rank.
Prove: only after elementary row operations.

9. We can conclude:

10. e.g. To figure out the rank of matrices

13. Elementary Matrices
Definition: An elementary matrix is one obtained by
performing a single elementary operation
on an identity matrix.
The followings are three different kinds of
elementary matrices:

14. They also include the ones obtained by performing a single elementary column operation on an identity matrix.

15. Properties of Elementary Matrices1.

17. The Relation Between Elementary Matrices and Elementary Operations.
Transposes of elementary matrices are same
to themselves. 2.
Elementary matrices are all invertible.
The Relation Between Elementary
Matrices and Elementary Operations.
Look at an example first.

19. Elementary row operations are equivalent to multiply an
elementary matrix on the left. And multiplying an elementary
matrix on the right stands for Elementary column operations.

20. e.g. To find the normal form of matrix A and use
elementary matrices to show the elementary operations
We can prove it

22. =？