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Gauss-Jordan elimination with partial pivoting reduces the augmented matrix to the identity, leaving the solution in the final column. Partial pivoting keeps the arithmetic numerically stable.
Gauss-Jordan
reduce [A | b] to [I | x] by row scaling and row subtraction, pivoting on the largest available entry
The reduction gives x = 1, y = 2, z = 3.
Dividing by a tiny pivot amplifies rounding error; choosing the largest available magnitude keeps the computation stable.