Row Echelon Form Calculator — Step-by-Step Solutions
The Row Echelon Form Calculator transforms any matrix into row echelon form (REF) or reduced row echelon form (RREF) using Gaussian elimination with partial pivoting. It shows every row operation step by step so you can follow the entire process.
Choose your matrix dimensions (2x2, 2x3, 3x3, 3x4, or 4x4), enter your values, and select REF, RREF, or step-by-step mode. The calculator displays the resulting matrix, the rank, and the pivot columns. The 3x4 option supports augmented matrices for solving systems of linear equations.
Enter matrix values
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How to Use the Row Echelon Form Calculator
- Choose a mode: Row Echelon Form (REF), Reduced Row Echelon Form (RREF), or Step-by-Step Operations.
- Select the matrix dimensions using the buttons: 2x2, 2x3, 3x3, 3x4 (augmented), or 4x4.
- Enter your matrix values in the grid. Empty cells are treated as zero.
- The calculator applies Gaussian elimination with partial pivoting and displays every row operation.
- Review the resulting matrix, rank, and pivot columns below the step-by-step working.
Formula and Theory
A matrix is in row echelon form (REF) when all zero rows are at the bottom, the leading entry (pivot) in each non-zero row is to the right of the pivot in the row above, and all entries below each pivot are zero.
A matrix is in reduced row echelon form (RREF) when it satisfies all REF conditions and additionally each pivot is 1 and is the only non-zero entry in its column.
Gaussian elimination transforms a matrix into REF by applying three elementary row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a scalar multiple of one row to another.
Gauss-Jordan elimination extends Gaussian elimination to produce RREF by also eliminating entries above each pivot. RREF is unique for any given matrix.
Partial pivoting selects the row with the largest absolute value in the pivot column to reduce numerical errors. The rank of the matrix equals the number of pivot positions.
Worked Examples
Example 1: RREF of a 2x2 matrix
Problem: Find the RREF of [[2, 4], [1, 3]].
Solution: Swap rows if needed, scale R1 by 1/2 to get [1, 2], then R2 = R2 - R1 gives [0, 1]. Back-substitute: R1 = R1 - 2*R2 gives [1, 0].
Answer: [[1, 0], [0, 1]]. Rank = 2.
Example 2: REF of a 3x3 matrix
Problem: Find the REF of [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Solution: R2 = R2 - 4*R1, R3 = R3 - 7*R1. Then scale and eliminate to get upper triangular form.
Answer: [[1, 2, 3], [0, 1, 2], [0, 0, 0]]. Rank = 2.
Example 3: Augmented matrix (3x4)
Problem: Solve x + y + z = 6, 2x + 3y + z = 14, x + 2y + 3z = 14 using RREF.
Solution: Form the augmented matrix [[1,1,1,6],[2,3,1,14],[1,2,3,14]] and reduce to RREF.
Answer: [[1,0,0,1],[0,1,0,2],[0,0,1,3]]. x=1, y=2, z=3.
Frequently Asked Questions
What is the difference between REF and RREF?
Is the row echelon form unique?
What does the rank tell me?
What is partial pivoting?
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