Gauss Jorden and Gauss Elimination method.pptx

linear algebra

1. Gauss Jorden and Gauss Elimination method To solve system of linear equations

2. Define and History Ali shan 19011598-044

3. System of linear equation If all equations in a system are linear, the system is a system of linear equation or linear system. Example: 2y + z = -8 x -2y -3z = -x+y+2z = 3 There are two methods to solve these equations: Direct and Iterative method

4. Gauss Elimination Define:- It is also known as  row reduction , is an  algorithm  in  linear algebra  for solving a  system of linear equations . It is usually understood as a sequence of operations performed on the corresponding  matrix  of coefficients. This method can also be used to find the  rank  of a matrix, to calculate the  determinant  of a matrix, and to calculate the inverse of an  invertible square matrix . History :- The method is named after  Carl Friedrich Gauss  (1777–1855). Some special cases of the method - albeit presented without proof - were known to  Chinese mathematicians  as early as circa 179 CE.

5. Gauss Jorden Elimination Gauss–Jordan elimination  to refer to the procedure which ends in reduced echelon form. The name is used because it is a variation of Gaussian elimination as described by  Wilhelm Jordan  in 1888 . It is a further calculation of gauss elimination method. For Guass elimination: [A]= For guass Jordan Elimination: [A]=  

6. Why we need these methods? It is usually understood as a sequence of operations performed on the corresponding  matrix  of coefficients. This method can also be used to find the  rank  of a matrix, to calculate the  determinant  of a matrix, and to calculate the inverse of an  invertible square matrix . Gaussian Elimination  helps to put a matrix in row echelon form, while  Gauss - Jordan Elimination  puts a matrix in reduced row echelon form. For small systems (or by hand), it is usually more convenient to use  Gauss - Jordan elimination  and explicitly solve for each variable represented  in the  matrix system.

7. Techniques of Gauss Elimination and GaussJorden method Rameen iftikhar 19011598-042

8. Guass Elimination method First consider the system of linear equations as; AX=B   a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 Find the augmented matric for the given system of equations as; C =[A;B] C=   Use row operation to transform the augmented matrix into the Row Echelon Form.  

9. An elementary row operation is;   Interchange the two rows. Multiply a row by non-zero constant. Add a multiple of the row to another row.   Now check the resulting matrix and re-interpret it as a system of linear equations. The resulting matrix will be;   Find the solution of the equations by interpreting the equations .  

10. Guass Jordan method First consider the system of linear equations as; AX=B a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 Find the augmented matric for the given system of equations as; C=[A;B] C= Use row operation to transform the augmented matrix into the Reduced Row Echelon Form.  

11. An elementary row operation is;   Interchange the two rows. Multiply a row by non-zero constant. Add a multiple of the row to another row.   Now check the resulting matrix and re-interpret it as a system of linear equations. The resulting matrix will be;   Find the solution of the equations .  

12. Examples of Gauss Elimination method AHSAN MEHBOOB 19011598-023

13. EXAMPLE.1: The equation is: x − y + z =8 x +3 y − z = − 2 3 x −2 y − 9 z =9 In augmented form = = − 2R1+R2=R2 and −3R1+R3=R3    

14. = Interchange  R2 and  R3 = −5R2+R3=R3 = − 1/57R3=R3 Using   back-substitution to get the values: Z=1 x − y+z=8 y − 12z= − 15 Hence The solution set is (4,-3,1)  

15. Example.2: The linear equation is: 2x+3y+2z=-3 x+y+z=0 -x+2y-3z=-1 Solution: In augmented form = = Interchanging R 1 and R 2  

16. = R 2 =-2R 1 +R 2 & R 3 =R 1 +R 3 = R 3 =-3R 2 +R 3 = R 3 =- R 3 Here z=-4 y=-3 x+y+z=0 X-3-4=0 X=7 S.S={(7,-3,-4)}  

17. Examples of Gauss Jordan Elimination method Amina zubair 19011598-035

18. Example.1: 1.Solve the system of linear equations by using Gauss-Jordan Eliminating Method: 2y + z = -8 x -2y -3z = 0 -x+y+2z = 3 Solution: The augmented matrix of the system is: Interchanging R 1 and R 2  

19. R 3 =R 3 +R 1 Interchanging R 2 and R 3 R 3 =R 3 +2R 2 R 1 =R 1 -2R 2 R 2 =R 2 -R 3 & R 1 =R 1 -R 3 Multiple R 2 &R 3 both by (-1) Therefore x=-4,y=-5&z=2. Thus the solution of the given equations is (-4,-5,2)  

20. Example.2: 2.Solve the system of linear equations by using Gauss-Jordan Eliminating Method: x + y + z = 2 6x -4y+5z = 31 5x+2y+2z=13 Solution: The augmented matrix of the system is : R 2 =R 2 -6R 1  

21. R 3 =R 3 -5R 1 R 2 =R 2 /- 10 R 1 =R 1 -R 2 R 3 =R 3 +3R 2 R 3 =-( 10/27)R 3  

22. R 1 =R 1 -( 9/10)R 3 R 2 =R 2 -( 1/10)R 3 Therefore x=3,y=-2&z=1 . Thus the solution of the given equations x + y + z = 2 6x -4y+5z = 31 5x+2y+2z=13 Are x=3 y=-2 z=1.  

23. Applications

24. Applications: Computing determinants By using row operations of Gaussian elimination we can find out the determinant of any square matrix Finding the inverse of a matrix A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix. Computing ranks and bases

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