Reduced Row Echelon Form Calculator
Reduced Row Echelon Form Calculator simplifies matrices using Gaussian elimination, producing unique RREF solutions for linear systems and rank analysis.
Enter a matrix in row format (use semicolon ; for new rows).
Example: 1,2,1; 2,4,0; -1,2,3
Equation Preview:
Result
What is Reduced Row Echelon Form Calculator?
A Reduced Row Echelon Form (RREF) Calculator is a tool designed to transform any given matrix into its reduced row echelon form using systematic row operations. The reduced row echelon form is a canonical matrix representation that uniquely solves linear systems, reveals matrix rank, and simplifies the study of linear independence. Each nonzero row has a leading 1, every leading 1 is the only nonzero entry in its column, and each leading 1 in successive rows moves to the right.
About the Reduced Row Echelon Form Calculator
The calculator automates Gaussian and Gauss–Jordan elimination, applying row operations:
- Swap two rows
- Multiply a row by a nonzero scalar
- Add a multiple of one row to another row
These steps reduce the matrix to RREF, which satisfies the following conditions:
For example, solving a system \(Ax=b\) is simplified when the augmented matrix \([A|b]\) is reduced to RREF, directly showing variable dependencies.
How to Use this Reduced Row Echelon Form Calculator
- Input the coefficients of your matrix (augmented if solving linear systems).
- Click "Calculate" — the tool applies Gauss–Jordan elimination automatically.
- View step-by-step transformations, from the original matrix to its final RREF.
- Interpret the result to identify solutions, rank, or dependencies in the system.
Example
Consider the system:
The augmented matrix is:
After row operations, the RREF becomes:
which corresponds to the unique solution \(x=2, y=1, z=-1\).
This Reduced Row Echelon Form Calculator provides accurate, step-by-step results, making it an essential tool for students and professionals in linear algebra.