Reduced Echelon Form Calculator

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Reduced Row Echelon Form (RREF) Calculator

Enter the dimensions of your matrix and fill in the values. This tool uses Gaussian elimination with partial pivoting to transform your matrix into its unique Reduced Row Echelon Form.

2 3 4 5
2 3 4 5 6

Resulting RREF Matrix:

What is Reduced Row Echelon Form (RREF)?

In linear algebra, a matrix is in Reduced Row Echelon Form (RREF) if it satisfies four specific conditions:

  • All non-zero rows are above any rows of all zeros.
  • The leading coefficient (the first non-zero number from the left, called a pivot) of a non-zero row is always 1.
  • Each leading 1 is to the right of the leading 1 in the row above it.
  • Each column containing a leading 1 has zeros everywhere else in that column.

How the Calculator Works

This calculator employs the Gauss-Jordan Elimination algorithm. It performs elementary row operations to simplify the matrix. These operations include:

  1. Row Swapping: Moving a row with a higher absolute value in the pivot position to the top to maintain numerical stability.
  2. Scalar Multiplication: Multiplying a row by a constant to turn the pivot element into a 1.
  3. Row Addition/Subtraction: Adding multiples of one row to another to create zeros in columns where a pivot exists.

Example Calculation

Suppose you have a system of linear equations represented by the following augmented matrix:

[ 2, 4, -2 | 8 ]
[ 4, 9, -3 | 14 ]
[ -2, -3, 7 | 10 ]

After applying row operations, the calculator will produce the RREF:

[ 1, 0, 0 | -1 ]
[ 0, 1, 0 | 2 ]
[ 0, 0, 1 | -1 ]

This result directly tells us the variables: x = -1, y = 2, and z = -1.

Why Use RREF?

RREF is essential for solving systems of linear equations, finding the rank of a matrix, calculating inverses, and determining the basis of a vector space. It provides the simplest representation of a matrix while preserving its row space.

function generateMatrixInputs() { var rows = parseInt(document.getElementById('matrixRows').value); var cols = parseInt(document.getElementById('matrixCols').value); var container = document.getElementById('matrixInputGrid'); var html = ''; for (var i = 0; i < rows; i++) { html += ''; for (var j = 0; j < cols; j++) { html += ''; } html += ''; } html += '
'; container.innerHTML = html; } function clearMatrix() { var inputs = document.getElementsByClassName('matrix-cell'); for (var i = 0; i < inputs.length; i++) { inputs[i].value = 0; } document.getElementById('rrefResultSection').style.display = 'none'; } function calculateRREF() { var rows = parseInt(document.getElementById('matrixRows').value); var cols = parseInt(document.getElementById('matrixCols').value); var matrix = []; for (var i = 0; i < rows; i++) { matrix[i] = []; for (var j = 0; j < cols; j++) { var val = parseFloat(document.getElementById('cell-' + i + '-' + j).value); if (isNaN(val)) val = 0; matrix[i][j] = val; } } var lead = 0; for (var r = 0; r < rows; r++) { if (cols <= lead) break; var i = r; while (matrix[i][lead] === 0) { i++; if (rows === i) { i = r; lead++; if (cols === lead) break; } } if (cols === lead) break; // Swap rows var temp = matrix[i]; matrix[i] = matrix[r]; matrix[r] = temp; // Normalize pivot var val = matrix[r][lead]; if (val !== 0) { for (var j = 0; j < cols; j++) { matrix[r][j] /= val; } } // Eliminate other rows for (var i = 0; i < rows; i++) { if (i !== r) { var val2 = matrix[i][lead]; for (var j = 0; j < cols; j++) { matrix[i][j] -= val2 * matrix[r][j]; } } } lead++; } displayMatrix(matrix); } function displayMatrix(matrix) { var output = document.getElementById('matrixOutput'); var resultSection = document.getElementById('rrefResultSection'); var html = ''; for (var i = 0; i < matrix.length; i++) { html += ''; for (var j = 0; j < matrix[i].length; j++) { // Format number to avoid floating point artifacts var num = matrix[i][j]; if (Math.abs(num) < 1e-10) num = 0; var formatted = Number.isInteger(num) ? num : num.toFixed(3).replace(/\.?0+$/, ""); html += ''; } html += ''; } html += '
' + formatted + '
'; output.innerHTML = html; resultSection.style.display = 'block'; // Smooth scroll to result resultSection.scrollIntoView({ behavior: 'smooth', block: 'nearest' }); } // Initial Generation generateMatrixInputs();

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