One more question based on Gauss Elimination Method

Que:-

           x + 2y + z    =3

           2x + 3y +3z = 10

            3x – y + 2z = 13

 Sol:-

[1        2        1]  [x]     [3]

[2        3        3]  [y]  = [10]

[3      -1         2]  [z]     [13]

[a₁₁     a₁₂       a₁₃]  [x]    [b₁]

[0       a₂₂        a₂₃]  [y] = [b₂]

[0       0           a₃₃]  [z]     [b₃]

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

R_i  -  a_i1 × R₁              for [ I = 2^nd & 3^rd ]

        a₁₁

For| i = 2|

               R₂ – a₂₁ × R_i

                      a₁₁

               R₂ – 2  ×  R₁

                      1

               R₂ – 2R₁

               a₂₁ = a₂₁ – a₁₁

                     = 2 – 2(1)

              |a₂₁= 0|

a₂₂ = a₂₂ – 2a₁₂

     

       = 3 – 2(2)

       = 3 – 4

|a₂₂= -1|

a₂₃ = a₂₃ – 2a₁₃

       = 3 – 2(1)

|a₂₃ = 1|

          

               b₂ = b₂ – 2b₁

                    = 10 – 2(3)

              |b₂= 4|

| i = 3 |

               R₃ – a₃₁ ×  R₁

                      a₁₁

                       R₃ – 3 × R₁

                          1

                R₃ – 3R₁

a₃₁ =  a₃₁ – 3a₁₁

       = 3 – 3(1)

|a₃₁= 0|

a₃₂= a₃₂ – 3a₁₂

      = -1 – 3(2)

          |a₃ = -7|

a₃₃= a₃₃ – 3a₁₃

      = 2 – 3(1)

    |a₃₁ = -1|

b₃ = b₃ – 3b₁

     = 13 – 3(3)

|b₃= 4|

[1        2        1]     [x]     [3]

[0       -1        1]    [y] =  [4]

[0       -7       -1]    [z]     [4]

To eliminate coefficient of y from 3^rd row.

| i = 3|

          R_i -  a_i2 × R₂

                 a₂₂

         R₃ – a₃₂ × R₂

                a₂₂

         R₃ – (-7) × R₂

                (-1)

         R₃ – 7R₂

a₃₂ = a₃₂ – 7a₂₂

      = -7 – 7(-1)

|a₃₂ = 0|

a₃₃ = a₃₃ – 7a₂₃

      = -1 – 7(1)

|a₃₃ = -8|

b₃ = b₃ – 7b₂

     = 4 – 7(4)

|b₃ = -24|

[1        2        1]  [x]      [3]  --------(i)

[0       -1        1]  [y] =  [4]  ---------(ii)

[0        0       -8]  [z]      [-24] -------(iii)

-8z = -24 

    z = 24

           8

       |z = 3|

-y + z = 4

-y = 4 – 3

  

 |y = -3|

x + 2y + z =3

x + 2(-1) + 3 = 3

         x = 3 + 2 – 3

|x = 2|

[x = 2     ,     y = -1     ,     z = 3]