Overview
Gauss-Jordan is an extension of Gauss elimination that reduces the augmented matrix all the way to reduced row echelon form (RREF). Instead of just upper triangular, you also eliminate entries above each pivot, giving the solution directly without back-substitution.
While Gauss elimination gets you halfway (upper triangular), Gauss-Jordan goes all the way — it turns the coefficient matrix into the identity matrix, so the answer column directly gives you each variable's value.
Gauss-Jordan Method
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Mathematical Foundation
Formal Statement
Required Conditions
- • Same as Gauss elimination — the system must be consistent
- • Additional row operations are needed to eliminate entries above pivots
- • The diagonal entries are all normalized to 1
Common Mistakes & Pitfalls
- ⚠ Not normalizing pivot rows to have 1 on the diagonal.
- ⚠ Only doing forward elimination (that's just Gauss, not Gauss-Jordan).
- ⚠ Sign errors when eliminating entries above pivots.