One unique solution
x₁ + x₂ = 3
x₁ − x₂ = 1
| 1 | 0 | 2 |
| 0 | 1 | 1 |
- x1 = 2
- x2 = 1
Every coefficient column has a pivot and the constants column does not. Rank of A equals rank of [A | b] equals the number of variables.
Linear algebra, made clearer
Solve the system. Understand the answer.
Enter coefficients and constants. See the reduced matrix, solution type, and free variables.
Calculated in your browser
02RREF result
Example resultReduced row echelon form
| 1 | 0 | 1 | 5/3 |
| 0 | 1 | −1 | 2/3 |
| 0 | 0 | 0 | 0 |
Infinitely many solutions
- x1 = 5/3 − 1·t1
- x2 = 2/3 + 1·t1
- x3 = t1
Each t is an arbitrary real parameter.
- Exact fractions
- Every operation shown
- Your matrix stays in this browser
The working, step by step.
5 operations · exact values01R2 ← R2 + (−2) R1
Matrix after step 1 1 2 −1 3 0 −3 3 −2 3 3 0 7 02R3 ← R3 + (−3) R1
Matrix after step 2 1 2 −1 3 0 −3 3 −2 0 −3 3 −2 03R2 ← (−1/3) R2
Matrix after step 3 1 2 −1 3 0 1 −1 2/3 0 −3 3 −2 04R1 ← R1 + (−2) R2
Matrix after step 4 1 0 1 5/3 0 1 −1 2/3 0 −3 3 −2 05R3 ← R3 + (3) R2
Matrix after step 5 1 0 1 5/3 0 1 −1 2/3 0 0 0 0
Reading the answer
How to read an augmented matrix RREF result.
An augmented matrix [A | b] stores a linear system. Each row is one equation, the columns before the divider hold the coefficients of x₁, x₂ and so on, and the last column holds the constants b. Row operations change the matrix but never the solution set, so the reduced matrix answers the original system.
Rank of A and rank of [A | b]
The rank of A counts pivots among the coefficient columns. The rank of [A | b] counts every pivot, including one in the constants column. If the two ranks differ, the system is inconsistent. If they match and equal the number of variables, the solution is unique. If they match but are smaller, the system has infinitely many solutions with one free variable for each coefficient column without a pivot.
Unique, infinite, or no solution
Infinitely many solutions
x₁ + 2x₂ − x₃ = 3
2x₁ + x₂ + x₃ = 4
3x₁ + 3x₂ = 7
| 1 | 0 | 1 | 5/3 |
| 0 | 1 | −1 | 2/3 |
| 0 | 0 | 0 | 0 |
- x1 = 5/3 − 1·t1
- x2 = 2/3 + 1·t1
- x3 = t1
Both ranks are 2, but there are 3 variables. Column 3 has no pivot, so x₃ becomes the free parameter t₁.
No solution
x₁ + x₂ = 2
x₁ + x₂ = 3
| 1 | 1 | 0 |
| 0 | 0 | 1 |
The last row reads 0 = 1. The constants column has a pivot, so rank of [A | b] is larger than rank of A.
Free variables and parametric form
A coefficient column without a pivot becomes a free variable. The augmented matrix calculator names these parameters t₁, t₂ and so on and writes every pivot variable in terms of them, which is the parametric form of the solution. The constants column is never a free variable.
Tips for entering a system
- Move every variable to the left side and every constant to the right before you enter the rows.
- Enter 0 for a variable that does not appear in an equation. Blank cells are reported as errors.
- Use one row per equation and keep the variables in the same column order in every row.
- Choose “Columns incl. b” as the number of variables plus one.
The why behind the answer
Make the next step make sense.
Augmented matrix calculator questions.
Does it show the RREF steps for an augmented matrix?
Yes. Every swap, scaling and row replacement is listed with the exact matrix it produces, including the constants column.
Can it solve a system with more equations than variables?
Yes. Enter up to 6 equations. Extra dependent equations become zero rows; a contradictory equation produces a row that reads 0 = nonzero and the calculator reports no solution.
Can it give the solution in vector form?
Not yet. Solutions are written as equations in free parameters, such as x₃ = t₁. Vector-form output is not part of this calculator.
What about an ordinary matrix that is not a system?
Use the RREF matrix calculator. It reduces any matrix without treating the last column as constants and also supports REF.