(2.8.1)--2.14ApplicationofElementaryOpera.ppt

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1、Linear AlgebraApplication of Elementary Operations and Elementary MatricesWe know that if A is invertible,then A1 is also invertible.From the above theorem,there exists a chain of elementary matrices P1,P2,Ps,such that P1P2Ps E=A1 (1)then P1P2Ps A=E (2)Hence from(1),there is a chain of elementary ro

2、w operations which transform E to A1,and from(2),the same chain of elementary row operations which transform A to E.Procedure for finding the inverse of an nn matrix A as follows:Step 1 Adjoin the columns of the nn identity matrix E to the matrix A to obtain the n2n matrix A E.Step 2 If possible,con

3、vert the first n columns of A E to the identity matrix E by using only elementary row operations.Step 3 Once the matrix A E is converted to E B,the matrix B will satisfy AB=E,that is mean that BA is also equal to E and B=A1.Example.Find the inverse ofSolution.In this case,we formand convert the firs

4、t three columns to 33 identity matrix.Solution.Our first objective is to convert to echelon form.This can be done by adding to the second row 2 times the first row and then adding to the third row 1/2 times the first row,to obtainSolution.Next,we interchange the second row and the third row to obtai

5、n an echelon formThe first three column are an echelon form,so detA0,and is therefore invertible.Solution.Our next objective is to convert leading entries to ones and to obtain zeros above the leading entries.This may be done by multiplying the third row by 1/2 to obtainSolution.We then add to the s

6、econd row 9/2 times the third row and add to the first row 1 times the third row to obtain Solution.Next,we multiply the second row by 2/5 to obtainand then add to the first row 3 times the second row to obtain Solution.Finally,we multiply the first row by 1/2 to obtainSolution.The final three colum

7、ns will now be the inverse of A,that is Since this final matrix is multiplied by A,the result is the identity matrix E.Note.In general,the least number of operations is used if we first convert to echelon form and then work from the bottom row upward,in turn convert each leading entry to one and the entries above those leading ones to zeros.