Row Echelon Form Calculator
Convert a 2x3 or 3x4 augmented matrix to row echelon form (REF) using Gaussian elimination. See every row operation, the final echelon form, and the solution obtained by back substitution.
Row echelon form (REF) is a simplified version of a matrix where every row starts with more leading zeros than the row above it. Gaussian elimination uses elementary row operations to reach this form.
Row Echelon Form Properties:
- All zero rows are at the bottom
- The leading entry (first non-zero number) of each non-zero row is to the right of the leading entry of the row above
- All entries below a leading entry are zero
Elementary Row Operations:
- Swap two rows
- Multiply a row by a non-zero scalar
- Add a multiple of one row to another
These operations do not change the solution set of the system.
Back Substitution:
Once the matrix is in REF, solve from the bottom row up. The last row gives the value of the last variable directly. Substitute back to find the others.
Example:
For the system x + y = 5, 2x + 3y = 13: Augmented matrix: [[1, 1, 5], [2, 3, 13]] After R2 = R2 - 2*R1: [[1, 1, 5], [0, 1, 3]] Solution: y = 3, x = 5 - 3 = 2