Solve any system of linear equations by reducing the augmented matrix [A|b] to RREF — every row operation shown step by step with exact fractions.
System Setup
Enter the augmented matrix [A | b]. Coefficients go in the white cells; the right-hand side b goes in the yellow cells. You can enter non-square systems too.
Solution
Step-by-Step Row Operations
Phase 1 — Forward elimination: Transform A to upper triangular (REF of augmented matrix). Phase 2 — Back substitution: Scale pivots to 1 and eliminate entries above each pivot (complete RREF).
Gauss-Jordan Method — Theory
Algorithm: Apply elementary row operations to the augmented matrix $[A|b]$ until it reaches RREF:
Forward pass (REF): For each column, find the pivot row, swap if needed, then use elimination to zero out all entries below the pivot.
Scale pivots: Divide each pivot row by its pivot value to make the pivot equal to 1.
Backward pass (RREF): Eliminate entries above each pivot.
Read solution: If RREF is consistent, read off the values. Free variables indicate infinite solutions.
Three possible outcomes:
• Unique solution — rank(A) = rank([A|b]) = n (number of variables).
• Infinite solutions — rank(A) = rank([A|b]) < n; free variables exist.
• No solution — rank(A) < rank([A|b]); a row of the form [0 0 … 0 | k], k ≠ 0 appears.
Gauss-Jordan elimination: solving Ax = b
Form the augmented matrix [A | b] and use row operations to reach reduced row echelon form. If rank(A) = rank([A | b]) = n, the system has a unique solution; if the ranks are equal but less than n, it has infinitely many solutions with free variables; if rank(A) < rank([A | b]), it has no solution (it is inconsistent).
Common questions
How do I know if a system is inconsistent?
A row of the form [0 0 … 0 | c] with c ≠ 0 appears in the reduced matrix.
What are free variables?
Variables whose columns have no pivot. They can take any value, which gives infinitely many solutions.