1. GGAUSS JORDAN NAME OF TEACHER MR:SAMAN NAME OF STUDENTS Salih Saman Hugir Jalal Rojha t Nojda r

2.  Abstract  Introduction  GAUSS JORDAN METHOD  HISTORY ABOUT GAUSS JORDAN METHOD  Some point about gauss Jordan  Here the steps to gauss Jordan elimination  Example  Refernce 1

3. Abstract We talk about GAUSS JORDAN METHOD and we talk about some way to solve these matrices and some example and we will know many thing about gauss Jordan method 2

4. Introductio n  For inverting a matrix, G auss -j ordan elimination is about as efficient as any other method. for solving sets of linear equations, G auss -j ordan elimination produces both the solution of the equations for one or more right – hand side vectors b,andalso the matrix inverse −1  however, its principal weaknesses are -(i) that it requires all the right – hand sides to be stored and manipulated at the same time, and -(ii) that when the inverse matrix is not desired, G auss -j ordan is three times slower than the best alternative technique for solving a single linear set  The method`s principal strength is that it is as stable as any other direct method, perhaps even a bit more stable when full 3

5. G AUSS J ORDAN METHOD Some authors use the term Gaussian elimination to refer only to the procedure until the matrix is in echelon form, and use the term Gauss-Jordan elimination to refer to the procedure which ends in reduced echelon form. In linear algebra, Gauss–Jordan elimination is analgorithm for getting matrices in reduced rowechelon form using elementary row operations. Itis a variation of Gaussian elimination. 4

6. H ISTORY ABOUT G AUSS J ORDAN METHOD it is a variation of Gaussian elimination asdescribed by Wilhelm Jordan in 1887.Wilhelm Jordan However, the method also appears in an articleby Clasen published in the same year. Jordanand Clasen probably discovered Gauss–Jordan elimination independently. 5

7. H ERE ARE THE point TO G AUSS -J ORDAN ELIMINATION : Turn the equations into an augmented matrix. Use elementary row operations on matrix [A|b] totransform A into diagonal form. Make sure there areno zeros in the diagonal. Divide the diagonal element and the right- handelement (of b) for that diagonal element's row sothat the diagonal element is equal to one. 6

8. H ERE ARE THE STEPS TO G AUSS - J ORDAN ELIMINATION : 1. Write the augmented matrix of the system. 2. Use row operations to transform the augmented matrix in the form described below, which is called the reduced row echelon form (RREF). (a)The rows (if any) consisting entirely of zeros are grouped together at the bottom of the matrix. (b)In each row that does not consist entirely of zeros, the leftmost nonzero element is a 1 (called aleading 1 or a pivot). 7

9. G AUSS - J ORDAN  Variation of gauss elimination  Primary motive for introducing this method is that it provides a simple and convenient method for computing the matrix inverse.  When an unknown is eliminated, it is eliminated from all other equations, rather than just the subsequent one  All rows are normalized by dividing them by their pivot elements  Elimination step results in an identity matrix 8

10. Example 1. Solve the following system byusing the Gauss-Jordan elimination method. x + y + z = 5 2x + 3y + 5z = 8 4x + 5z = 2 Solution: The augmented matrix of the system is the following. 111|5 235|8 405|2 We will now perform row operations until we obtain a matrix in reduced row echelon form. 111|5111|5 235|8013|− 2− 2 405|2405|2 9

11. 111|5111|5 R3-4R1R3-4R1 111|5111|5 R3+4R2 (1/13)R3 111|5111|5 R3+4R2 R1-R3 10

12. From this final matrix, we can read the solution of the system. It is R1-R2 11

13. Example 2: − − − Sol : − → ( → ( − − ) →− − − ) →− ( ) ( ) → ( → ( − ) − ) − → → (→ → ( ) −→ (−→ ( ) The answer is x=1, y=1, and z=2 12

14. Refern ce 1https://en.m.wilibooks.org/wiki/linearAlgebra/gass- jordanReductionhttps://en.m.wilibooks.org/wiki/linearAlgebra/gass- jordanReduction 2Internet 3Timothy Gowers; June Barrow-Green; Imre Leader (8 September 2008). The Princeton Companion to Mathematics. Princeton University Press. p. 607. ISBN 978-0-691-11880-2.The Princeton Companion to MathematicsISBN 13

15. 14