Gauss Elimination Method

a₁₁x₁  +  a₁₂x₂ +  a₁₃ x₃      = b₁

a₂₁x₁  +  a₂₂x₂ + a₂₃x₃        =b₂

a₃₁x₁  + a₃₂x₂  + a₃₃x₃        =b₃

[a₁₁         a₁₂         a₁₃]   [x₁]        [b₁]

[a₂₁             a₂₂       a₂₃]   [x₂]    =  [b₂]

[a₃₁         a₃₂         a₃₃]   [x₃]        [b₃]

To  eliminate coefficient of x₁ from 2^nd and 3^rd Row Recursive formula :-

[ R_i – a_i1 × R_i                    for [ I = 2 & 3]

        a₁₁

[a₁₁        a₁₂          a₁₃]  [x₁]         [b₁]

[a₂₁        a₂₂          a₂₃]  [x₂]    =   [b₂]

[a₃₁        a₃₂          a₃₃]  [x₃]          [b₃]

To eliminate coefficients of x₂ from 3^rd row

[R_i – a_i2 × R₂]

       a₂₂

Ques:-

           2x₁  +  2x₂  +  2x₃  =  14

              x₁  +  6x₂   -    x₃  =  13

             2x₁  -   x₂   +  2x₃  =   5

Sol:-

To eliminate coefficient of x₁ from 2^nd and 3^rd row.

For |i=2|

            R₂ – a₂₁ ×  R₁

                   a₁₁

  a₂₁ = a₂₁ – 1 × a₁₁

                      2

         = 1 – 1 × 2

                 2

|a₂₁=0|

a₂₂= a₂₂ – a₂₁ × a₁₂

               a₁₁

   = 6 – 1 × 8

           2

|a₂₂ = 2|

a₂₃= a₂₃ × a₂₁  × a₁₃

               a₁₁

      = - 1 – 1 × 2

                2

|a₂₃=- 2|

[2      8        2] [x₁]        [14]

[0       2     -2]  [x₂]   =   [13]

[2      -1      2]  [x₃]        [5]

For |i = 3|

        R₃ – a₃₁ × R₁

              a₁₁

a₃₁ = a₃₁ –  a₃₁ × a₁₁

                 a₁₁

      = 2 – 2 × 2

              2

|a₃₁ = 0|

a₃₂ = a₃₂ – a₃₁ × a₁₂

                a₁₁

       = - 1 – (-1) × 8

                   2

|a₃₂= -9|

a₃₃ = a₃₃ –  a₃₁ × a₁₃

                 a₁₁

       = 2 – 2  × 2

               2

|a₃₃=0|

b₃ = b₃ – a₃₁ × b₁  => 5 – 2 ×14

             a_{11                  2} 

                     

     = 5 – 14

|b₃= -9|

[2       8        2]  [x₁]        [14]

[0       2       -2]  [x₂]   =   [ 6 ]

[0      -9        6]  [x₃]        [-9]

To eliminate x₂ from 3^rd row.

| R_i –  a_i2 × R₂ |

          a₂₂

for |i = 3|

       R₃ –  a₃₂ × R₂

              a₂₂

 a₃₂ = a₃₂ –  a₃₂ × a₂₂

                  a₂₂

        = - 9 – (-9) × 2

                   2

|a₃₂=0|

a₃₃ = a₃₃ –  a₃₂ × a₂₃

                 a₂₂

      = 0 – (-9) ×(-2)

                2

|a₃₃= - 9|

b₃ = b₃ – a₃₂  × b₂

             a₂₂

     = - 9 – (-9) × 6

                 2

     = -9 – (-27)

|b₃= 18|

[2       8        2]  [x₁]       [14]

[0       2       -2]  [x₂]  =   [ 6 ]

[0       0       -9]  [x₃]       [18]

-       9x₃ = 18

  x₃ = - 18

            9

|x₃= 2|

  2x₂ + (2x₃)  = 6

  2x₂ – 2x₃     = 6

    x₂ – x₃       = 3

           x₂      = 3 – 2

|x₂ = 1|

2x₁   +   8x₂   +   2x₃   =   14

2x₁   +  8(1)  +   2(-2)=   14

2x₁   =  14 – 8 + 4

   x₁  = 10

            2

|x₁ = 5|

[x₁ = 5   ,   x₂ = 1   ,   x₃ = -2 ]