Reduced Row Echelon Form (RREF) Calculator

Perform Gauss-Jordan elimination on any matrix to find its simplest form.

Rows:

Columns:

Resulting Matrix

'; container.innerHTML = html; document.getElementById('rrefResultSection').style.display = 'none'; } function calculateRREF() { var rows = parseInt(document.getElementById('matrixRows').value); var cols = parseInt(document.getElementById('matrixCols').value); var matrix = []; // Read values 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)) { document.getElementById('errorMessage').innerText = "Please fill all cells with valid numbers."; document.getElementById('errorMessage').style.display = "block"; return; } matrix[i][j] = val; } } document.getElementById('errorMessage').style.display = "none"; // Gauss-Jordan Algorithm 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 i and r var temp = matrix[i]; matrix[i] = matrix[r]; matrix[r] = temp; // Divide row r by matrix[r][lead] to make lead 1 var val = matrix[r][lead]; if (val !== 0) { for (var j = 0; j < cols; j++) { matrix[r][j] /= val; } } 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++; } // Clean up small numbers (floating point precision) for (var r = 0; r < rows; r++) { for (var c = 0; c < cols; c++) { if (Math.abs(matrix[r][c]) < 1e-10) matrix[r][c] = 0; // Format to fixed decimal to avoid long strings matrix[r][c] = parseFloat(matrix[r][c].toFixed(4)); } } // Display result var resultHtml = ''; for (var r = 0; r < rows; r++) { resultHtml += ''; for (var c = 0; c < cols; c++) { resultHtml += ''; } resultHtml += ''; } resultHtml += '

' + matrix[r][c] + '

'; document.getElementById('resultMatrixContainer').innerHTML = resultHtml; document.getElementById('rrefResultSection').style.display = 'block'; } // Initialize grid on load window.onload = generateMatrixGrid;

Understanding Reduced Row Echelon Form (RREF)

In linear algebra, the Reduced Row Echelon Form (RREF) is a specific state of a matrix achieved through a sequence of elementary row operations. This process, known as Gauss-Jordan elimination, is essential for solving systems of linear equations, finding the rank of a matrix, and calculating inverse matrices.

Conditions for RREF

A matrix is in Reduced Row Echelon Form if it satisfies the following four conditions:

All rows consisting entirely of zeros are at the bottom of the matrix.
In every non-zero row, the first non-zero entry (the leading coefficient or pivot) is a 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 in all its other entries.

How to Use the RREF Calculator

Select Dimensions: Enter the number of rows and columns for your matrix and click "Generate Grid".
Input Coefficients: Enter the numerical values of your matrix or system of equations into the input boxes.
Calculate: Click the "Calculate Reduced Row Echelon Form" button.
Analyze Result: The tool will output the simplified matrix, showing the pivots and zero-filled columns.

Practical Example

Consider the following 3×3 matrix:

            [ 1, 2, 3 ]

            [ 4, 5, 6 ]

            [ 7, 8, 9 ]

After applying the RREF algorithm, the resulting matrix becomes:

            [ 1, 0, -1 ]

            [ 0, 1, 2 ]

            [ 0, 0, 0 ]

This result shows that the matrix is singular (rank 2) because it contains a row of zeros, indicating that the original rows were linearly dependent.

Frequently Asked Questions

What is the difference between REF and RREF?
Row Echelon Form (REF) only requires zeros below the pivots. Reduced Row Echelon Form (RREF) requires zeros both above and below the pivots, and all pivots must be 1.

Can any matrix be put into RREF?
Yes, every matrix has a unique Reduced Row Echelon Form, regardless of the sequence of row operations used to get there.

Why do I get decimals in the result?
Calculations involve division by pivots. If the division isn't clean (e.g., 1/3), the resulting matrix will contain decimal representations.

Reduced Row Echelon Form Calculator