Use this calculator to find the Reduced Row Echelon Form (RREF) of a given matrix. Enter the dimensions (rows and columns) of your matrix, then input the individual elements, and the calculator will perform Gaussian elimination to provide the RREF.
// Helper function to create a matrix filled with zeros
function createMatrix(rows, cols) {
var matrix = [];
for (var i = 0; i < rows; i++) {
matrix.push(new Array(cols).fill(0));
}
return matrix;
}
// Helper function to get matrix values from input fields
function getMatrixFromInputs(rows, cols, prefix) {
var matrix = createMatrix(rows, cols);
for (var i = 0; i < rows; i++) {
for (var j = 0; j < cols; j++) {
var inputId = prefix + (i + 1) + (j + 1);
var inputElement = document.getElementById(inputId);
if (!inputElement) {
// This can happen if dimensions change after generating fields
return null;
}
var value = parseFloat(inputElement.value);
if (isNaN(value)) {
return null; // Indicate invalid input
}
matrix[i][j] = value;
}
}
return matrix;
}
// Helper function to format a matrix for display
function formatMatrixOutput(matrix) {
if (!matrix || matrix.length === 0) return "Invalid matrix.";
var output = "
";
for (var i = 0; i < matrix.length; i++) {
output += "
";
for (var j = 0; j < matrix[0].length; j++) {
// Format numbers to a reasonable precision, handle -0
var val = matrix[i][j];
if (Math.abs(val) < 1e-9) val = 0; // Treat very small numbers as 0
output += "
" + val.toFixed(4) + "
";
}
output += "
";
}
output += "
";
return output;
}
// Main function to convert a matrix to Reduced Row Echelon Form (RREF)
function toReducedRowEchelonForm(matrix) {
var rows = matrix.length;
if (rows === 0) return [];
var cols = matrix[0].length;
if (cols === 0) return [];
// Create a deep copy to avoid modifying the original input matrix
var m = matrix.map(function(row) { return row.slice(); });
var lead = 0;
for (var r = 0; r = cols) {
break;
}
var i = r;
while (i < rows && m[i][lead] === 0) {
i++;
}
if (i < rows) {
// Swap rows
var temp = m[r];
m[r] = m[i];
m[i] = temp;
// Divide row r by m[r][lead] to make the pivot 1
var lv = m[r][lead];
if (lv !== 0) { // Avoid division by zero if somehow pivot became zero
for (var j = 0; j < cols; j++) {
m[r][j] /= lv;
}
}
// Eliminate other entries in the current column (make them 0)
for (var i2 = 0; i2 < rows; i2++) {
if (i2 !== r) {
var lv2 = m[i2][lead];
for (var j2 = 0; j2 < cols; j2++) {
m[i2][j2] -= lv2 * m[r][j2];
}
}
}
}
lead++;
}
return m;
}
// Function to perform the RREF calculation
function calculateRREF() {
var rows = parseInt(document.getElementById("matrixRows").value);
var cols = parseInt(document.getElementById("matrixCols").value);
if (isNaN(rows) || isNaN(cols) || rows <= 0 || cols <= 0) {
document.getElementById("result").innerHTML = "Please enter valid positive dimensions for the matrix.";
return;
}
var matrix = getMatrixFromInputs(rows, cols, "m");
if (matrix === null) {
document.getElementById("result").innerHTML = "Please enter valid numbers for all matrix elements.";
return;
}
var rrefMatrix = toReducedRowEchelonForm(matrix);
document.getElementById("result").innerHTML = "
Reduced Row Echelon Form (RREF):
" + formatMatrixOutput(rrefMatrix);
}
// Function to dynamically generate matrix input fields based on dimensions
function generateMatrixInputs() {
var rows = parseInt(document.getElementById("matrixRows").value);
var cols = parseInt(document.getElementById("matrixCols").value);
var matrixInputDiv = document.getElementById("matrixInputFields");
if (isNaN(rows) || isNaN(cols) || rows <= 0 || cols 10 || cols > 10) {
matrixInputDiv.innerHTML = "Please enter valid positive dimensions (max 10×10 for practical use).";
document.getElementById("result").innerHTML = ""; // Clear previous result
return;
}
var html = "
Enter Matrix Elements:
";
html += "
";
for (var i = 0; i < rows; i++) {
html += "
";
for (var j = 0; j < cols; j++) {
// Preserve existing values if fields are regenerated
var currentVal = document.getElementById("m" + (i + 1) + (j + 1)) ? document.getElementById("m" + (i + 1) + (j + 1)).value : "0";
html += "
";
}
html += "
";
}
html += "
";
html += "";
matrixInputDiv.innerHTML = html;
document.getElementById("result").innerHTML = ""; // Clear previous result
}
// Initialize matrix inputs on page load
window.onload = generateMatrixInputs;
Understanding Row Echelon Form (REF) and Reduced Row Echelon Form (RREF)
In linear algebra, matrices are fundamental tools for representing and solving systems of linear equations. Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) are standardized forms that simplify matrices, making them easier to analyze and solve.
What is Row Echelon Form (REF)?
A matrix is in Row Echelon Form if it satisfies the following conditions:
All non-zero rows are above any rows of all zeros.
The leading entry (the first non-zero number from the left, also called the pivot) of each non-zero row is to the right of the leading entry of the row immediately above it.
All entries in a column below a leading entry are zeros.
It's important to note that the Row Echelon Form of a matrix is not unique; different sequences of row operations can lead to different REF matrices for the same initial matrix.
What is Reduced Row Echelon Form (RREF)?
Reduced Row Echelon Form is a more stringent form of REF. A matrix is in RREF if it satisfies all the conditions for REF, plus two additional conditions:
The leading entry in each non-zero row is 1 (called a leading 1).
Each column containing a leading 1 has zeros everywhere else (above and below the leading 1).
Unlike REF, the Reduced Row Echelon Form of a matrix is unique. This uniqueness makes RREF particularly useful for solving systems of linear equations, finding the rank of a matrix, and determining the inverse of a matrix.
How the Calculator Works
This calculator uses the Gauss-Jordan elimination method to transform your input matrix into its unique Reduced Row Echelon Form. The process involves a series of elementary row operations:
Swapping two rows: Interchanging the positions of any two rows.
Multiplying a row by a non-zero scalar: Scaling all elements in a row by a constant.
Adding a multiple of one row to another row: Replacing a row with the sum of itself and a multiple of another row.
These operations are applied systematically to achieve the RREF, where leading entries become 1s and all other entries in their respective columns become 0s.
Example: Calculating RREF
Let's consider the following 3×3 matrix:
1
2
-1
2
2
4
3
5
-2
If you input these values into the calculator (Rows: 3, Columns: 3), the calculator will perform the necessary row operations to yield its Reduced Row Echelon Form. The RREF for this matrix is:
1.0000
0.0000
5.0000
0.0000
1.0000
-3.0000
0.0000
0.0000
0.0000
This RREF indicates that the original system of equations represented by this matrix has infinitely many solutions, as the last row of zeros implies a free variable.