Row Echelon Form Calculator
Enter the coefficients of your 3×3 matrix to calculate its Row Echelon Form (REF) using Gaussian elimination.
Resulting Row Echelon Form:
What is Row Echelon Form?
In linear algebra, a matrix is in Row Echelon Form (REF) if it satisfies three specific conditions resulting from elementary row operations:
- All non-zero rows are above any rows of all zeros.
- The leading coefficient (or pivot) of a non-zero row is always to the right of the leading coefficient of the row above it.
- All entries in a column below a leading coefficient are zeros.
How to Calculate Echelon Form
The process of converting a matrix to echelon form is called Gaussian Elimination. It involves three primary operations:
- Swapping: Interchanging two rows.
- Scaling: Multiplying a row by a non-zero constant.
- Pivoting: Adding a multiple of one row to another row to eliminate variables.
Example Calculation:
If you have a matrix where Row 2 is exactly double Row 1, the elimination process will turn Row 2 into zeros. For instance, if Row 1 is [1, 2, 3] and Row 2 is [2, 4, 6], performing
If you have a matrix where Row 2 is exactly double Row 1, the elimination process will turn Row 2 into zeros. For instance, if Row 1 is [1, 2, 3] and Row 2 is [2, 4, 6], performing
R2 - 2R1 results in [0, 0, 0].
Why is this important?
Converting a matrix to echelon form is the first step in solving systems of linear equations. It allows mathematicians and engineers to determine the Rank of a matrix, calculate the determinant, and find the inverse of a matrix. It is a fundamental tool in data science, physics simulations, and structural engineering.
Rank of a Matrix
The rank is defined as the number of non-zero rows in the Row Echelon Form. This indicates the number of linearly independent rows or columns in the matrix, which tells us if a system of equations has a unique solution, infinitely many solutions, or no solution at all.