Reduced Row Echelon Calculator

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Reduced Row Echelon Form (RREF) Calculator

Input your matrix coefficients below to transform it into Reduced Row Echelon Form using Gauss-Jordan elimination.

Resulting Matrix:

What is Reduced Row Echelon Form?

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

  • All non-zero rows are above any rows of all zeros.
  • The leading coefficient (the first non-zero number from the left, called the 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 that contains a leading 1 has zeros everywhere else in that column.

Steps for Gauss-Jordan Elimination

This calculator uses the Gauss-Jordan elimination method, which involves three types of elementary row operations:

  1. Swapping: Interchange two rows.
  2. Scaling: Multiply a row by a non-zero constant.
  3. Pivoting: Add a multiple of one row to another row.

Example Calculation

Suppose you have the following system of linear equations:

x + 2y + 3z = 9
2x – y + z = 8
3x + 0y – z = 3

By entering these coefficients (1, 2, 3, 9; 2, -1, 1, 8; 3, 0, -1, 3) into the calculator, the RREF process will solve for x, y, and z simultaneously.

function calculateRREF() { var matrix = []; var rows = 3; var cols = 4; // Load data from inputs for (var i = 0; i < rows; i++) { matrix[i] = []; for (var j = 0; j < cols; j++) { var val = parseFloat(document.getElementById('m' + i + j).value); if (isNaN(val)) { alert("Please enter valid numbers in all fields."); return; } 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 lead var lv = matrix[r][lead]; if (lv !== 0) { for (var j = 0; j < cols; j++) { matrix[r][j] /= lv; } } // Eliminate other rows for (var i = 0; i < rows; i++) { if (i !== r) { var lv2 = matrix[i][lead]; for (var j = 0; j < cols; j++) { matrix[i][j] -= lv2 * matrix[r][j]; } } } lead++; } displayResult(matrix); } function displayResult(matrix) { var resultDiv = document.getElementById('rref-result'); var matrixDisplay = document.getElementById('matrix-display'); var solutionSummary = document.getElementById('solution-summary'); resultDiv.style.display = 'block'; var html = ''; for (var i = 0; i < matrix.length; i++) { html += ''; for (var j = 0; j < matrix[i].length; j++) { // Fix floating point noise like -0.00000000000001 var val = matrix[i][j]; if (Math.abs(val) < 1e-10) val = 0; html += ''; } html += ''; } html += '
' + val.toFixed(2).replace(/\.00$/, "") + '
'; matrixDisplay.innerHTML = html; // Check if it's a solved 3×3 system (common use case) var x = matrix[0][3].toFixed(2).replace(/\.00$/, ""); var y = matrix[1][3].toFixed(2).replace(/\.00$/, ""); var z = matrix[2][3].toFixed(2).replace(/\.00$/, ""); if (matrix[0][0] == 1 && matrix[1][1] == 1 && matrix[2][2] == 1) { solutionSummary.innerHTML = "System Solution: x = " + x + ", y = " + y + ", z = " + z; } else { solutionSummary.innerHTML = "Process complete. The matrix is now in RREF."; } resultDiv.scrollIntoView({ behavior: 'smooth', block: 'nearest' }); } function clearMatrix() { for (var i = 0; i < 3; i++) { for (var j = 0; j < 4; j++) { document.getElementById('m' + i + j).value = 0; } } document.getElementById('rref-result').style.display = 'none'; }

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