Gaussian Elimination Calculator
Row-reduce any matrix to echelon and reduced row-echelon form with pivot tracking.
About Gaussian Elimination Calculator
Gaussian elimination transforms a matrix into reduced row echelon form (RREF) using elementary row operations: swapping rows, scaling rows, and adding multiples of one row to another. Partial pivoting is used for numerical stability. The pivot columns identify the linearly independent columns.
Formula
Row Operations
R_i ← R_i − (R_i[k]/R_pivot[k]) × R_pivot
- R_i
- Current row
- R_pivot
- Pivot row
- k
- Pivot column
Worked example
With the default values below, Gaussian Elimination Calculator returns Reduced Row Echelon Form of 1 0 0 2 0 1 0 3 0 0 1 -1. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Matrix A (rows separated by newlines or semicolons; values by commas or spaces)
- 2 1 -1 8 -3 -1 2 -11 -2 1 2 -3
Results
- Reduced Row Echelon Form
- 1 0 0 2 0 1 0 3 0 0 1 -1
- Rank
- 3
- Pivot Columns (1-indexed)
- 1, 2, 3
Step by step
- 1Original Matrix= 2 1 -1 8 -3 -1 2 -11 -2 1 2 -3
- 2After row reduction (RREF)= 1 0 0 2 0 1 0 3 0 0 1 -1
- 3Pivot columns= 1, 2, 3
- 4Rank= 3
Assumptions and limits
- The result is derived only from the single input listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the difference between REF and RREF?
Row Echelon Form has leading 1s with zeros below each pivot. Reduced Row Echelon Form additionally has zeros above each pivot, making the solution directly readable.
Can I solve a system of equations with this?
Yes. Enter the augmented matrix [A|b] where the last column is the constants vector. After RREF, the solution is directly readable from the last column.