Gaussian Elimination Calculator
Row-reduce any matrix to echelon and reduced row-echelon form with pivot tracking.
Inputs
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
Original Matrix
= 2 1 -1 8 -3 -1 2 -11 -2 1 2 -3
After row reduction (RREF)
= 1 0 0 2 0 1 0 3 0 0 1 -1
Pivot columns
= 1, 2, 3
Rank
= 3
How it works
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
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.