RREF Calculator
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Reduced row echelon form (RREF) is the simplest form a matrix can reach with row operations: every pivot is 1 and is the only nonzero entry in its column. The augmented matrix of x + 2y − z = 3, 2x − y + z = 4, 3x + y + 2z = 11 reduces to show x = 9/5, y = 8/5, z = 2.
| Matrix facts | |
| Size | 3 × 4 |
| Rank | 3 |
| Pivot columns | 1, 2, 3 |
| Row operations used | 8 |
| System of equations | |
| Solution | One solution |
Show steps
- Start with \(\left[\begin{array}{ccc|c}1 & 2 & -1 & 3 \\ 2 & -1 & 1 & 4 \\ 3 & 1 & 2 & 11\end{array}\right]\)
- Clear the rest of column 1: \(R_{2} \to R_{2} – 2R_{1},\quad R_{3} \to R_{3} – 3R_{1}\) \(\Rightarrow \left[\begin{array}{ccc|c}1 & 2 & -1 & 3 \\ 0 & -5 & 3 & -2 \\ 0 & -5 & 5 & 2\end{array}\right]\)
- Make the pivot 1: \(R_{2} \to \left(-\frac{1}{5}\right) R_{2}\) \(\Rightarrow \left[\begin{array}{ccc|c}1 & 2 & -1 & 3 \\ 0 & 1 & -\frac{3}{5} & \frac{2}{5} \\ 0 & -5 & 5 & 2\end{array}\right]\)
- Clear the rest of column 2: \(R_{1} \to R_{1} – 2R_{2},\quad R_{3} \to R_{3} + 5R_{2}\) \(\Rightarrow \left[\begin{array}{ccc|c}1 & 0 & \frac{1}{5} & \frac{11}{5} \\ 0 & 1 & -\frac{3}{5} & \frac{2}{5} \\ 0 & 0 & 2 & 4\end{array}\right]\)
- Make the pivot 1: \(R_{3} \to \tfrac{1}{2} R_{3}\) \(\Rightarrow \left[\begin{array}{ccc|c}1 & 0 & \frac{1}{5} & \frac{11}{5} \\ 0 & 1 & -\frac{3}{5} & \frac{2}{5} \\ 0 & 0 & 1 & 2\end{array}\right]\)
- Clear the rest of column 3: \(R_{1} \to R_{1} – \frac{1}{5}R_{3},\quad R_{2} \to R_{2} + \frac{3}{5}R_{3}\) \(\Rightarrow \left[\begin{array}{ccc|c}1 & 0 & 0 & \frac{9}{5} \\ 0 & 1 & 0 & \frac{8}{5} \\ 0 & 0 & 1 & 2\end{array}\right]\)
How to use the RREF calculator
- Type the matrix with one row per line and spaces between entries.
- Choose Augmented if the last column holds the constants of a system of equations, or Plain matrix otherwise.
- Press Row reduce. Every row operation is listed with the matrix after it, in exact fractions.
What reduced row echelon form means
A matrix is in RREF when:
- Any rows of all zeros are at the bottom.
- The first nonzero entry of each row (the pivot) is 1.
- Each pivot is to the right of the pivot in the row above.
- Each pivot is the only nonzero entry in its column.
Every matrix has exactly one RREF, so it is a reliable way to compare answers.
The three row operations (Gauss–Jordan elimination)
- Swap two rows: \(R_{i} \leftrightarrow R_{j}\)
- Multiply a row by a nonzero number: \(R_{i} \to kR_{i}\)
- Add a multiple of one row to another: \(R_{i} \to R_{i} + kR_{j}\)
Work column by column: make the pivot 1, then clear every other entry in that column.
Worked example
\(\left[\begin{array}{cc|c}1 & 2 & 5 \\ 3 & 4 & 6\end{array}\right]\). Use \(R_{2} \to R_{2} – 3R_{1}\) to get \(\left[\begin{array}{cc|c}1 & 2 & 5 \\ 0 & -2 & -9\end{array}\right]\). Then \(R_{2} \to -\tfrac{1}{2}R_{2}\) and \(R_{1} \to R_{1} – 2R_{2}\) give \(\left[\begin{array}{cc|c}1 & 0 & -4 \\ 0 & 1 & \frac{9}{2}\end{array}\right]\), so x = −4 and y = 9/2.
Reading the result
- One solution: every variable column has a pivot.
- Infinitely many solutions: some variable column has no pivot; that variable is free.
- No solution: a row reads 0 = nonzero, such as [0 0 | 1].
- Rank = number of pivots. A square matrix is invertible exactly when its RREF is the identity matrix.
Row reduction (RREF) practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Multiplying by 0 destroys information; the scale factor must be nonzero.
Quiz complete
Frequently asked questions
What is the difference between REF and RREF?
Row echelon form (REF) only needs zeros below each pivot. Reduced row echelon form also needs each pivot to be 1 and zeros above it, which makes the solution readable directly.
How do I solve a system with RREF?
Write the augmented matrix [A | b], reduce it, then read each pivot row as an equation. Columns without a pivot are free variables.
Can I enter fractions and decimals?
Yes. Entries like 3/4, -2/3 and 0.25 are converted to exact fractions, so the steps never show rounding errors.
What is the rank of a matrix?
The number of pivots in its RREF. It tells you how many independent rows (and columns) the matrix has.
Does the order of row operations matter?
Different sequences of operations can be used, but they always end at the same RREF.
Sources
- Paul’s Online Math Notes: Augmented Matrices
- OpenStax College Algebra 2e: Solving Systems with Gaussian Elimination
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