Gauss elimination/Jordan with pivoting

Gauss-Jordan method with partial pivoting: at every stage the largest element (in absolute value) of the pivot column is moved to the pivot position by a row swap, the pivot row is divided by the pivot to make it 1, and the other entries of that column (above and below) are made 0. The final matrix is the identity, so the right column is the solution.

Step 1: System

2x + 3y + 2z = 2
10x + 3y + 4z = 16
3x + 6y + z = 6

Step 2: Reduce to the identity form

Augmented matrix [A | b]:

 2.0000   3.0000   2.0000  |   2.0000
10.0000   3.0000   4.0000  |  16.0000
 3.0000   6.0000   1.0000  |   6.0000

Column 1: largest pivot is in row 2, so swap R1 ↔ R2:

10.0000   3.0000   4.0000  |  16.0000
 2.0000   3.0000   2.0000  |   2.0000
 3.0000   6.0000   1.0000  |   6.0000

R1 ← R1 / 10.0000:

1.0000  0.3000  0.4000  |  1.6000
2.0000  3.0000  2.0000  |  2.0000
3.0000  6.0000  1.0000  |  6.0000

R2 ← R2 − (2.0000) R1; R3 ← R3 − (3.0000) R1:

 1.0000   0.3000   0.4000  |   1.6000
 0.0000   2.4000   1.2000  |  -1.2000
 0.0000   5.1000  -0.2000  |   1.2000

Column 2: largest pivot is in row 3, so swap R2 ↔ R3:

 1.0000   0.3000   0.4000  |   1.6000
 0.0000   5.1000  -0.2000  |   1.2000
 0.0000   2.4000   1.2000  |  -1.2000

R2 ← R2 / 5.1000:

 1.0000   0.3000   0.4000  |   1.6000
 0.0000   1.0000  -0.0392  |   0.2353
 0.0000   2.4000   1.2000  |  -1.2000

R1 ← R1 − (0.3000) R2; R3 ← R3 − (2.4000) R2:

 1.0000   0.0000   0.4118  |   1.5294
 0.0000   1.0000  -0.0392  |   0.2353
 0.0000   0.0000   1.2941  |  -1.7647

R3 ← R3 / 1.2941:

 1.0000   0.0000   0.4118  |   1.5294
 0.0000   1.0000  -0.0392  |   0.2353
 0.0000   0.0000   1.0000  |  -1.3636

R1 ← R1 − (0.4118) R3; R2 ← R2 − (-0.0392) R3:

 1.0000   0.0000   0.0000  |   2.0909
 0.0000   1.0000   0.0000  |   0.1818
 0.0000   0.0000   1.0000  |  -1.3636

Step 3: Solution

From the final matrix (identity on the left), the unknowns are read directly:

x = 2.0909
y = 0.1818
z = -1.3636

Answer: x = 2.0909, y = 0.1818, z = -1.3636