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**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.

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**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 .

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**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.

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**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.

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**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 :

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**e.g.: Compute the rank of Matrix A.**

We can figure out the rank of the matrix by elementary operations

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

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We can conclude:

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

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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:

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

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**Properties of Elementary Matrices**

1.

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**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.

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**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.

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**e.g. To find the normal form of matrix A and use **

elementary matrices to show the elementary operations We can prove it

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