Gauss jordan method.pptx

Gauss jordan method

1. G AUSS J O R DAN METH O D  Some a u th o rs use the t e rm Ga u ssian eli m ination to ref e r o n ly to the pr o ce d ure u n til the matrix is in e ch e lon form, a n d use t h e term Gau ss -Jordan el i mination to refer to the proc e d u re which en d s in reduc e d e c helon for m .  In line a r alg e bra, Ga u s s – Jord a n e lim i n a tion is an alg o rithm for g e tting matrices in re d uc e d r ow echel o n form using ele m ent a ry r ow o p era t io n s. It is a v a riation of Ga u ssian eliminatio n .

2. H IS T O R Y ABOUT G AUSS J ORDAN METHOD  it is a v a riation of Ga u ssian elimination d e scribed by Wi l h e lm Jordan in 1 8 8 7. as  Howev e r , t h e me t h o d a lso a p p e a rs in a n article by Clas e n publish e d in t h e s a me ye a r . J o rdan a n d C lasen pro b a b ly discov e red Gaus s – Jord a n eli m ination inde p en d entl y .

3. H ERE ARE THE STE P S T O G AUS S - J ORD A N ELIMIN A TION :  T urn the eq u ati o ns i n to an au g mented matri x .  Use e l ementary row operations on mat r ix [ A | b] to t r ansform A i nto d i ag o nal fo r m. Make s u re t here are no zeros in the d i a g o n a l .  D i vi d e the diag o nal e l ement and the r i gh t - hand e l eme n t (of b) f or that d i a g o n a l e l eme n t's row so that the d i ag o nal e l ement is equ a l to o ne.

4.  Ex a mple 1 . So lv e t h e f o l l ow i ng sys t e m by u s i n g t h e Ga u s s - J o r d an e l imi n at i on met h o d . x + y + 2x + 3y 4x + 5 z z = 5 + 5z = 8 = 2

5. 5 2  Solution : fo l l o win g . The augmented m atrix of the system is the 1 2 4 1 3 1 |5 5 |8 5 |2  W e w i ll now p erf o rm row op e rat i ons u n til we o bt a in a matr i x in r e du c ed row ec h el o n form. 1 2 4 1 3 1 |5 5 |8 5 |2 1 1 1 4 5 1 3 -2

6. R 3-4 R 1 1 1 1 5 1 3 -2 4 5 2 1 1 1 5 1 3 -2 -4 1 -18

7. R 3 +4R2 1 1 1 5 1 3 -2 -4 1 -18 1 1 1 5 1 3 -2 13 -26

8. (1/ 1 3) R 3 1 1 1 5 1 3 -2 13 -26 1 1 1 5 1 3 -2 1 -2

9. R2-3R3 1 1 1 5 1 3 -2 1 -2 1 1 1 5 1 -2 1 4

10. 1 1 1 5 1 4 1 -2 1 4 1 -2 1 1 7 R 1- R 3

11. 1 1 7 1 4 1 -2 1 4 1 -2 1 3 R1 -R2

12.  F r om t hi s fi n a l mat ri x, s yst e m. It is we c an r e a d t h e s o l u ti on of t he X= 3 Y=4 Z=-2  Substitute x , y, z in system equation, if the right side = left side in three equation then the sol. is correct 3+4-2=5 2*3+3*4+5*(-2)=8 4*3+5*(-2)=2

13. Thank you