Matrix Echelon Form Calculator
body {
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background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 0;
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margin: 30px auto;
padding: 25px;
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.input-section, .output-section, .article-section {
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border-radius: 4px;
font-size: 1em;
flex-grow: 1; /* Allow inputs to grow */
min-width: 120px; /* Minimum width for inputs */
margin-bottom: 5px; /* Spacing when wrapping */
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width: 100%;
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#result {
background-color: #e8f5e9; /* Light green */
color: #1b5e20; /* Dark green for text */
padding: 15px;
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font-size: 1.2em;
text-align: center;
font-weight: bold;
margin-top: 20px;
border: 1px solid #a5d6a7;
white-space: pre-wrap; /* Preserve formatting for matrix output */
word-break: break-all; /* Break long lines */
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margin-bottom: 8px;
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margin-bottom: 10px;
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}
Matrix Echelon Form Calculator
Result: Echelon Form
Enter matrix dimensions and elements, then click calculate.
Understanding Matrix Echelon Forms
Matrices are fundamental mathematical objects used extensively in linear algebra, computer graphics, engineering, economics, and many other scientific fields. A key operation performed on matrices is transforming them into an "echelon form." This process simplifies the matrix, making it easier to solve systems of linear equations, determine rank, find the null space, and perform other advanced matrix operations.
What is Echelon Form?
There are two primary types of echelon forms:
- Row Echelon Form (REF): A matrix is in row echelon form if it satisfies the following conditions:
- All rows consisting entirely of zeros are at the bottom of the matrix.
- For each non-zero row, the first non-zero element (called the leading entry or pivot) is strictly to the right of the leading entry of the row above it.
- All entries in a column below a leading entry are zeros.
- Reduced Row Echelon Form (RREF): A matrix is in reduced row echelon form if it is in row echelon form and additionally satisfies:
- Every leading entry is 1.
- Each column containing a leading entry has zeros not only below but also above the leading entry.
How is Echelon Form Calculated?
The process of converting a matrix to echelon form involves applying a sequence of elementary row operations. These operations do not change the solution set of the system of linear equations represented by the matrix. The allowed operations are:
- Swapping two rows: Exchanging the positions of two rows.
- Scaling a row: Multiplying all elements of a row by a non-zero scalar.
- Adding a multiple of one row to another row: Adding a scalar multiple of one row to another row.
The calculator above uses these operations to systematically eliminate entries below the leading entries (for REF) and then zeros above the leading entries and makes leading entries 1 (for RREF). The most common algorithm used is Gaussian elimination (for REF) and Gauss-Jordan elimination (for RREF).
Use Cases
- Solving Systems of Linear Equations: Echelon forms provide a clear path to finding unique solutions, infinite solutions, or no solutions.
- Determining Matrix Rank: The rank of a matrix is equal to the number of non-zero rows in its echelon form.
- Finding Null Space: The null space (or kernel) of a matrix can be readily determined from its RREF.
- Matrix Inversion: For square matrices, RREF can be used to find the inverse.
- Linear Independence: Determining if a set of vectors is linearly independent.
This calculator automates these complex steps, providing a reliable tool for students, researchers, and professionals working with linear algebra.
// Global array to store matrix values
var matrix = [];
// Function to update the matrix input fields based on dimensions
function updateMatrixInputs() {
var rows = parseInt(document.getElementById("rows").value);
var cols = parseInt(document.getElementById("cols").value);
var container = document.getElementById("matrixInputsContainer");
container.innerHTML = "; // Clear previous inputs
matrix = []; // Reset matrix data
if (isNaN(rows) || isNaN(cols) || rows < 1 || cols < 1) {
document.getElementById("errorMessage").innerText = "Please enter valid positive numbers for rows and columns.";
return;
}
document.getElementById("errorMessage").innerText = ""; // Clear error
for (var i = 0; i < rows; i++) {
matrix[i] = [];
for (var j = 0; j < cols; j++) {
var input = document.createElement("input");
input.type = "text"; // Use text to allow for fractions if needed, though numbers are preferred
input.id = "r" + i + "c" + j;
input.value = "0"; // Default value
input.setAttribute("data-row", i);
input.setAttribute("data-col", j);
input.oninput = function() {
var r = parseInt(this.getAttribute("data-row"));
var c = parseInt(this.getAttribute("data-col"));
var val = parseFloat(this.value);
if (isNaN(val)) {
// Optionally handle non-numeric input, or var parseFloat handle it
// For now, we'll just var it be NaN or the invalid string
}
matrix[r][c] = val;
};
container.appendChild(input);
}
}
// Populate the global matrix array with initial values from the generated inputs
populateMatrixData();
}
// Function to read values from inputs and populate the 'matrix' array
function populateMatrixData() {
var rows = parseInt(document.getElementById("rows").value);
var cols = parseInt(document.getElementById("cols").value);
for (var i = 0; i < rows; i++) {
matrix[i] = [];
for (var j = 0; j < cols; j++) {
var inputElement = document.getElementById("r" + i + "c" + j);
if (inputElement) {
var val = parseFloat(inputElement.value);
matrix[i][j] = isNaN(val) ? 0 : val; // Use 0 if input is not a valid number
} else {
matrix[i][j] = 0; // Fallback if input element doesn't exist
}
}
}
}
// Function to display the matrix in a readable format
function formatMatrix(mat) {
if (!mat || mat.length === 0) return "[]";
var output = "";
var maxLen = 0;
// Find max length for alignment (optional, for cleaner output)
for (var i = 0; i < mat.length; i++) {
for (var j = 0; j maxLen) maxLen = s.length;
}
}
for (var i = 0; i < mat.length; i++) {
output += "[";
for (var j = 0; j < mat[i].length; j++) {
var valStr = mat[i][j].toFixed(4).toString();
// Padding for alignment
output += valStr.padStart(maxLen + 2, ' ');
if (j < mat[i].length – 1) {
output += ", ";
}
}
output += "]\n";
}
return output.trim();
}
// Main function to calculate Echelon Form (Gaussian Elimination)
function calculateEchelonForm() {
populateMatrixData(); // Ensure matrix array is up-to-date
var rows = parseInt(document.getElementById("rows").value);
var cols = parseInt(document.getElementById("cols").value);
var resultDiv = document.getElementById("result");
var errorDiv = document.getElementById("errorMessage");
errorDiv.innerText = ""; // Clear previous errors
if (isNaN(rows) || isNaN(cols) || rows < 1 || cols < 1) {
errorDiv.innerText = "Invalid matrix dimensions.";
resultDiv.innerText = "Please enter valid dimensions.";
return;
}
// Create a deep copy of the matrix to avoid modifying the original input values directly
var echelonMatrix = matrix.map(function(row) {
return row.slice();
});
var lead = 0; // Current column we are working on
var pivotRow = 0; // Current row we are using as a pivot
// Gaussian elimination process
for (var r = 0; r = cols) break; // Moved past the last column
var i = r; // Start searching for pivot from the current row 'r'
// Find a row with a non-zero entry in the current leading column
while (echelonMatrix[i][lead] === 0) {
i++;
if (i === rows) { // If we reached the end of the rows
i = r; // Reset to current row
lead++; // Move to the next column
if (lead === cols) {
// If we've run out of columns, we're done with this part
lead = cols; // Ensure loop breaks
break;
}
}
}
if (lead === cols) break; // Break outer loop if no more columns
// Swap rows if necessary to bring the pivot row to the current position 'r'
if (i !== r) {
var temp = echelonMatrix[i];
echelonMatrix[i] = echelonMatrix[r];
echelonMatrix[r] = temp;
}
// Normalize the pivot row (make the leading entry 1) – This is for RREF, but useful for REF too.
// For strict REF, this step can be skipped, but it doesn't hurt and standardizes pivots.
var pivotValue = echelonMatrix[r][lead];
if (pivotValue !== 0) { // Avoid division by zero
for (var j = lead; j < cols; j++) {
echelonMatrix[r][j] /= pivotValue;
}
}
// Eliminate other entries in the current leading column (below the pivot)
for (var i = 0; i < rows; i++) {
if (i !== r) { // Skip the pivot row itself
var factor = echelonMatrix[i][lead];
if (factor !== 0) { // Only perform if there's something to eliminate
for (var j = lead; j < cols; j++) {
echelonMatrix[i][j] -= factor * echelonMatrix[r][j];
}
}
}
}
lead++; // Move to the next column for the next pivot
}
// Clean up very small numbers close to zero due to floating point inaccuracies
for (var r = 0; r < rows; r++) {
for (var c = 0; c < cols; c++) {
if (Math.abs(echelonMatrix[r][c]) < 1e-10) {
echelonMatrix[r][c] = 0;
}
}
}
resultDiv.innerText = "Row Echelon Form:\n" + formatMatrix(echelonMatrix);
// — Optional: Calculate Reduced Row Echelon Form (Gauss-Jordan Elimination) —
// The loop above already performs steps similar to RREF by eliminating above and below,
// and normalizing the pivot. The key difference is the order and ensuring pivots are 1.
// The current implementation results in RREF if pivots are correctly handled.
// To be explicitly RREF, one might need another pass or refine the loop.
// For clarity, let's ensure pivots are 1 and all entries above pivots are 0.
// The previous loop might have already achieved this, but a final pass can enforce it strictly.
var rrefMatrix = matrix.map(function(row) { return row.slice(); }); // Start fresh for RREF demo
lead = 0;
for (var r = 0; r = cols) break;
var i = r;
while (rrefMatrix[i][lead] === 0) {
i++;
if (i === rows) {
i = r;
lead++;
if (lead === cols) break;
}
}
if (lead === cols) break;
// Swap rows
var temp = rrefMatrix[i];
rrefMatrix[i] = rrefMatrix[r];
rrefMatrix[r] = temp;
// Normalize pivot row
var pivotValue = rrefMatrix[r][lead];
if (pivotValue !== 0) {
for (var j = lead; j < cols; j++) {
rrefMatrix[r][j] /= pivotValue;
}
}
// Eliminate entries above and below the pivot
for (var i = 0; i < rows; i++) {
if (i !== r) {
var factor = rrefMatrix[i][lead];
for (var j = lead; j < cols; j++) {
rrefMatrix[i][j] -= factor * rrefMatrix[r][j];
}
}
}
lead++;
}
// Clean up small numbers
for (var r = 0; r < rows; r++) {
for (var c = 0; c < cols; c++) {
if (Math.abs(rrefMatrix[r][c]) < 1e-10) {
rrefMatrix[r][c] = 0;
}
}
}
resultDiv.innerText += "\n\nReduced Row Echelon Form (RREF):\n" + formatMatrix(rrefMatrix);
}
// Initialize the matrix input fields when the page loads
document.addEventListener("DOMContentLoaded", updateMatrixInputs);