What is the RREF Calculator?
This calculator transforms any matrix (up to 6 rows by 6 columns) into its Reduced Row Echelon Form (RREF) using Gauss-Jordan elimination. RREF is a unique canonical form of a matrix that makes it easy to solve linear systems, find the rank, identify pivot and free variables, and determine linear independence. The tool also reports the rank, which equals the number of nonzero (pivot) rows in the RREF.
How to Use It
Set the number of rows and columns, then type the matrix entries one row per line, separating numbers with spaces or commas. For an augmented system [A | b], simply include the constant column as the last column. Missing or blank entries are treated as zero. Click calculate to see the fully reduced matrix and its rank.
The Method Explained
Gauss-Jordan elimination processes columns left to right. For each pivot position it finds a row with a nonzero entry, swaps it into place, scales that row so the pivot becomes 1, and then subtracts multiples of the pivot row from every other row so the rest of the pivot column becomes 0. A matrix is in RREF when each leading entry is 1, is the only nonzero entry in its column, and appears to the right of the leading entry in the row above.
$$\text{RREF}\left( A_{\,m \times n} \right) \;\xrightarrow{\text{Gauss-Jordan}}\; R$$The reduction is governed by the following row operations:
$$\begin{gathered} \text{RREF}\left( A_{\,m \times n} \right) \;\xrightarrow{\text{Gauss-Jordan}}\; R \\[1.5em] \text{where}\quad \left\{ \begin{aligned} R_{r,\,j} &\to \frac{R_{r,\,j}}{R_{r,\,\text{lead}}} \quad (\text{normalize pivot to }1) \\ R_{k,\,j} &\to R_{k,\,j} - R_{k,\,\text{lead}}\, R_{r,\,j} \quad (k \neq r) \\ m &= \text{Rows}, \quad n = \text{Columns} \end{aligned} \right. \end{gathered}$$
Worked Example
Take the matrix \([[1, 2, 3], [4, 5, 6]]\). Subtract 4 times row 1 from row 2 to get \([[1, 2, 3], [0, -3, -6]]\). Scale row 2 by \(-1/3\): \([[1, 2, 3], [0, 1, 2]]\). Subtract 2 times row 2 from row 1: \([[1, 0, -1], [0, 1, 2]]\). This is the RREF, and the rank is \(2\).
FAQ
What is the difference between REF and RREF? Row echelon form (REF) only requires zeros below each pivot, while RREF additionally requires leading 1s and zeros above each pivot, making the form unique.
How do I find the rank? The rank is the number of nonzero rows after reduction, shown at the top of the result.
Can I solve a system of equations? Yes. Enter the augmented matrix [A | b]; the RREF directly gives the solution or shows whether the system is inconsistent or has free variables.
