Presentation on theme: "Elementary Operations of Matrix"— Presentation transcript:
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 areall called elementary operationsFunctionHow to transform a matrix into an echelon one.
4 Equivalent Matrix normal form of A If matrix A can be transformed into B by someelementary operations, we say A and B are equivalent,and we marked as A B.Equivalent Matrices have the following characters：normal form of ATheory: any matrixhas its ownnormal form.
5 Corollary:matrices A and B are equivalent if As last e.g.：Corollary:matrices A and B are equivalent ifand 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.
17 The Relation Between Elementary Matrices and Elementary Operations. Transposes of elementary matrices are sameto themselves.2.Elementary matrices are all invertible.The Relation Between ElementaryMatrices and Elementary Operations.Look at an example first.