Reduced Row Echelon Form (RREF) Calculator
Convert any matrix to reduced row echelon form, see each row operation, identify pivots, find rank, and interpret augmented matrices as systems of equations.
Background
Reduced row echelon form, or RREF, is a simplified matrix form used to solve systems of linear equations, identify pivot columns, determine rank, and understand whether a system has one solution, infinitely many solutions, or no solution.
How to use this RREF Calculator
- Choose the number of rows and columns in your matrix.
- Enter values directly into the matrix grid, or paste a full matrix into the paste box.
- Choose Augmented matrix if the last column represents constants in a system of equations.
- Click Calculate RREF to see the reduced row echelon form, pivot columns, rank, and row-operation steps.
- Use the interpretation section to check whether an augmented system has one solution, infinitely many solutions, or no solution.
How this calculator works
- The calculator searches each column for a nonzero pivot entry.
- If needed, it swaps rows to move a pivot into the correct row.
- It scales the pivot row so the pivot becomes 1.
- It eliminates all other entries in the pivot column.
- The process continues until the matrix is in reduced row echelon form.
Formula & Concepts Used
Reduced row echelon form: every leading entry is 1, each pivot column has zeros above and below the pivot, and pivot positions move to the right as you go down the rows.
Elementary row operations: swap two rows, multiply a row by a nonzero constant, or add a multiple of one row to another row.
Pivot column: a column that contains a leading 1 in the RREF.
Rank: the number of pivot columns in the matrix.
Free variable: a variable whose column does not contain a pivot in an augmented system.
No solution test: an augmented matrix is inconsistent if a row becomes 0 0 0 | nonzero.
Example Problems & Step-by-Step Solutions
Example 1: System with one solution
Row-reduce the augmented matrix:
[1 2 −1 | 3][2 4 1 | 9]
[−1 −2 2 | −2]
After row reduction, every variable column has a pivot. That means the system has exactly one solution.
Example 2: System with infinitely many solutions
Row-reduce the augmented matrix:
[1 2 −1 | 3][2 4 −2 | 6]
[3 6 −3 | 9]
The rows are dependent, so at least one variable becomes free. Because there is no contradiction, the system has infinitely many solutions.
Example 3: System with no solution
Row-reduce the augmented matrix:
[1 2 −1 | 3][2 4 −2 | 8]
[3 6 −3 | 9]
If the RREF contains a row like 0 0 0 | 1, the system is inconsistent and has no solution.
FAQs
What is reduced row echelon form?
Reduced row echelon form is a simplified matrix form where each pivot is 1, each pivot column has zeros everywhere else, and pivots move to the right as you move down the matrix.
What is RREF used for?
RREF is used to solve systems of linear equations, find pivot columns, determine matrix rank, identify free variables, and check whether a system is consistent.
How do I know if a system has no solution?
An augmented system has no solution if its RREF contains a row where all coefficient entries are zero but the constant entry is nonzero.
How do I know if a system has infinitely many solutions?
A consistent augmented system has infinitely many solutions when at least one variable column does not contain a pivot, creating one or more free variables.
Does this calculator use exact fractions?
Yes. The calculator uses exact fraction arithmetic internally, which helps avoid decimal rounding errors during row reduction.
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