Enter a coefficient or augmented matrix and choose where to stop. Gaussian elimination produces row echelon form, while Gauss-Jordan continues until every pivot column is reduced.
Result
The augmented system has one solution
RREF(A)=[10015759]
Conditions
The last column is interpreted as the constants column.
Pivots are chosen with exact arithmetic.
The two variable columns contain pivots, so the solution is unique.
Steps
Eliminate below the first pivot Replace row 2 by row 2 minus twice row 1.[102−55−9]
Scale the second pivot Divide row 2 by negative 5.[1021559]
Clear above the second pivot Replace row 1 by row 1 minus twice row 2.[10015759]
✓
Independent check
Substitution gives 7/5+2(9/5)=5 and 2(7/5)-9/5=1, matching both original equations.
Scope
What this row reduction covers
Gaussian elimination uses elementary row operations to reach row echelon form (REF), then solves by back-substitution. Gauss-Jordan elimination continues to reduced row echelon form (RREF), where each pivot is the only nonzero entry in its column.
Stop at row echelon form
Use Gaussian elimination to create pivots with zeros below them, then preserve REF for back-substitution.
Examples: upper triangular REF, back-substitution
Continue to reduced row echelon form
Use Gauss-Jordan elimination to make every pivot 1 and clear all other entries in each pivot column.
Examples: unique RREF, free-variable RREF
Solve augmented linear systems
Interpret the last column as constants and distinguish one solution, infinitely many solutions, and no solution.
Examples: pivot in every variable column, free variable, inconsistent row
Reduce an ordinary matrix
Treat every column as matrix data when no augmented separator is selected, useful for rank and pivot analysis.
Examples: rank, pivot columns, linear dependence
How to use it
Enter enough information for one clear task
1
Choose coefficient or augmented input
Use an augmented matrix for a system of equations and identify the last column as constants. Use an ordinary matrix for rank or pivot structure.
2
Choose REF or RREF
Select Gaussian elimination to stop at REF or Gauss-Jordan elimination to continue to the unique RREF.
3
Enter every value exactly
Use integers, finite decimals, or fractions. Exact fractions avoid rounding a pivot or residual to zero.
4
Interpret the pivot pattern
Read pivot and free-variable columns, and check for an inconsistent row of the form [0 ... 0 | c] with c nonzero.
Worked inputs
Examples to try
Use these examples to recognize the method, compare equivalent forms, and check your own work.
Unique two-variable system
Use two pivots and clear both pivot columns.
[122−151]
Expected result
[10015759]
Infinite solutions
The second row reduces to zero, leaving one free variable.
[122436]
Expected result
x=3−2t,y=t
Inconsistent system
Elimination creates the impossible equation 0=1.
[121225]
Expected result
[101021]⇒∅
REF without clearing above pivots
Eliminate entries below successive pivots and stop before backward clearing.
12−1250142
Expected result
10021012−1
RREF of a dependent matrix
Reduce until pivot columns have zeros above and below their leading ones.
121241361
Expected result
100010−120
Row swap for a zero leading entry
Swap the rows before using the first-column pivot.
[012−141]
Expected result
[100132]
Fractional pivot
Keep one-third exact after elimination.
[211−110]
Expected result
[10013131]
Complete example
Solve a three-variable system and verify every equation
The augmented matrix has three variable columns. Reaching a pivot in each one will produce a unique solution.
12−11−121136312
1
Clear below the first pivot
Use row 1 to eliminate the first entries of rows 2 and 3.
1001−331−146−918
2
Create the third pivot
Add row 2 to row 3, producing a simple equation for z.
1001−301−136−99
3
Reduce upward
Scale the pivots and clear entries above them to reach RREF.
100010001123
x=1,y=2,z=3
Verification: Substitution gives 1+2+3=6, 2-2+3=3, and -1+4+9=12, so all three original equations hold.
Avoidable errors
Common mistakes and how to fix them
Calling every echelon form RREF
Problem: Stop when entries below pivots are zero and label the matrix reduced row echelon form.
Why it matters: RREF also requires each pivot to be 1 and the only nonzero entry in its column.
Better approach: Label the intermediate result REF, or continue backward clearing for RREF.
Changing only part of a row
Problem: Apply a row replacement to coefficients but not to the augmented constant.
Why it matters: That changes the represented equation and therefore the solution set.
Better approach: Apply every row operation across all columns, including the augmented column.
Treating a free variable as zero
Problem: Set every nonpivot variable to zero and report only one vector.
Why it matters: A free variable represents a family of solutions, not a fixed value unless the problem imposes one.
Better approach: Introduce a parameter and express each pivot variable in terms of it.
Missing an inconsistent row
Problem: Ignore [0 0 | 1] and continue back-substitution.
Why it matters: That row represents 0=1, so no vector satisfies the full system.
Better approach: Check every reduced row for a nonzero constant after all coefficient entries become zero.
Trust the result for the right reasons
Checks, assumptions, and limits
How results are checked
Every displayed elementary row operation is replayed on the complete row.
The result is tested against the formal REF or RREF conditions selected.
System solutions are substituted into every original equation.
Rank, pivot, free-variable, and inconsistency conclusions are derived from the final matrix.
When to stop and revise the input
Matrices are limited to four rows and four columns, including an augmented column.
Entries must be finite real integers, decimals, or simple fractions.
Symbolic parameters and complex entries are outside this release.
The page does not infer whether the last column is augmented unless that mode is selected.
Common questions
Reduce a matrix to REF or RREF FAQ
What is the difference between Gaussian and Gauss-Jordan elimination?
Gaussian elimination stops at REF and usually uses back-substitution. Gauss-Jordan continues until every pivot column is reduced, producing RREF directly.
Is row echelon form unique?
No. Different valid row-operation sequences can produce different REF matrices. RREF is unique for a given matrix.
How does RREF show infinitely many solutions?
At least one variable column lacks a pivot and no inconsistent row appears. The nonpivot variables become free parameters.
How does row reduction show no solution?
An augmented row with all zero coefficients and a nonzero constant represents an impossible equation such as 0=1.
Do row operations change the solution set?
Swapping rows, multiplying a row by a nonzero number, or adding a multiple of one row to another produces an equivalent system with the same solutions.
Sources and curriculum alignment
This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.
