Use this calculator to find the Row Echelon Form (REF) of any given matrix. Enter the number of rows and columns, then input the individual matrix elements. The calculator will perform Gaussian elimination to transform your matrix into its row echelon form.
Input Matrix:
Resulting Row Echelon Form:
// Function to generate matrix input fields
function generateMatrixInputs() {
var numRows = parseInt(document.getElementById('numRows').value);
var numCols = parseInt(document.getElementById('numCols').value);
var container = document.getElementById('matrixInputContainer');
var calculateButton = document.getElementById('calculateButton');
if (isNaN(numRows) || numRows < 1 || isNaN(numCols) || numCols < 1) {
container.innerHTML = 'Please enter valid numbers for rows and columns (minimum 1).';
calculateButton.style.display = 'none';
return;
}
var tableHTML = '
';
for (var i = 0; i < numRows; i++) {
tableHTML += '
';
container.innerHTML = tableHTML;
calculateButton.style.display = 'block'; // Show calculate button once inputs are generated
document.getElementById('outputMatrixContainer').innerHTML = "; // Clear previous output
}
// Function to perform Gaussian elimination for Row Echelon Form
function calculateEchelonForm() {
var numRows = parseInt(document.getElementById('numRows').value);
var numCols = parseInt(document.getElementById('numCols').value);
var matrix = [];
// Read matrix values from input fields
for (var i = 0; i < numRows; i++) {
var row = [];
for (var j = 0; j < numCols; j++) {
var inputId = 'matrix_' + i + '_' + j;
var value = parseFloat(document.getElementById(inputId).value);
if (isNaN(value)) {
document.getElementById('outputMatrixContainer').innerHTML = 'Please enter valid numbers for all matrix elements.';
return;
}
row.push(value);
}
matrix.push(row);
}
// Gaussian Elimination for Row Echelon Form
var lead = 0;
var epsilon = 1e-9; // For floating point comparisons
for (var r = 0; r = numCols) {
break;
}
var i = r;
while (i < numRows && Math.abs(matrix[i][lead]) epsilon) {
for (var j = lead; j < numCols; j++) {
matrix[r][j] /= lv;
}
}
// Eliminate elements below pivot
for (var k = r + 1; k < numRows; k++) {
var factor = matrix[k][lead];
for (var j = lead; j < numCols; j++) {
matrix[k][j] -= factor * matrix[r][j];
}
}
lead++;
}
displayMatrix(matrix, 'outputMatrixContainer');
}
// Function to display a matrix in a table
function displayMatrix(matrix, targetId) {
var container = document.getElementById(targetId);
var tableHTML = '
';
for (var i = 0; i < matrix.length; i++) {
tableHTML += '
';
for (var j = 0; j < matrix[0].length; j++) {
// Format numbers to a reasonable precision
tableHTML += '
The Row Echelon Form (REF) of a matrix is a fundamental concept in linear algebra, providing a simplified, standardized representation of any given matrix. It's a crucial step in solving systems of linear equations, determining the rank of a matrix, and understanding the properties of linear transformations.
What is Row Echelon Form?
A matrix is in Row Echelon Form if it satisfies the following three conditions:
All non-zero rows are above any rows of all zeros. This means if there are any rows consisting entirely of zeros, they must be at the bottom of the matrix.
The leading entry (also called the pivot) of each non-zero row is 1. The leading entry is the first non-zero number in a row, read from left to right.
Each leading entry is in a column to the right of the leading entry of the row above it. This creates a "staircase" pattern of leading ones, moving down and to the right.
All entries in a column below a leading entry are zero. This ensures that the leading one is the only non-zero entry in its column below its row.
It's important to distinguish Row Echelon Form (REF) from Reduced Row Echelon Form (RREF). RREF has an additional condition: all entries in a column above a leading entry are also zero. REF only requires zeros below the leading entry.
Why is Row Echelon Form Important?
Solving Systems of Linear Equations: Converting the augmented matrix of a system of linear equations into REF allows for straightforward back-substitution to find the solutions.
Determining Matrix Rank: The rank of a matrix is equal to the number of non-zero rows in its row echelon form.
Finding Basis Vectors: The pivot columns in the original matrix (corresponding to the leading ones in REF) form a basis for the column space of the matrix.
Simplifying Complex Matrices: REF provides a simpler, more manageable form of a matrix while preserving its essential linear algebraic properties.
The Process: Gaussian Elimination
The method used to transform a matrix into its Row Echelon Form is called Gaussian elimination. It involves a series of elementary row operations:
Swapping two rows: This allows you to bring a non-zero entry to a pivot position.
Multiplying a row by a non-zero scalar: This is used to make a leading entry equal to 1.
Adding a multiple of one row to another row: This is used to create zeros below the leading entries.
The process systematically works from the top-left of the matrix, creating leading ones and then using them to eliminate entries below them, moving column by column until the REF conditions are met.
Example of Row Echelon Form Transformation
Consider the following matrix:
[ 1 2 3 ]
[ 4 5 6 ]
[ 7 8 9 ]
Using Gaussian elimination, we would perform the following steps (simplified):
The first leading entry is already 1.
Eliminate the entries below the first leading 1:
R2 = R2 – 4*R1
R3 = R3 – 7*R1
This would result in:
[ 1 2 3 ]
[ 0 -3 -6 ]
[ 0 -6 -12 ]
Make the new leading entry in the second row 1:
R2 = R2 / -3
This would result in:
[ 1 2 3 ]
[ 0 1 2 ]
[ 0 -6 -12]
Eliminate the entry below the second leading 1:
R3 = R3 + 6*R2
This would result in the Row Echelon Form:
[ 1 2 3 ]
[ 0 1 2 ]
[ 0 0 0 ]
This calculator automates this complex process, allowing you to quickly find the Row Echelon Form for any matrix you input, making it a valuable tool for students and professionals alike.