Augmented Matrix Calculator
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
line-height: 1.6;
margin: 0;
padding: 20px;
background-color: #f8f9fa;
color: #333;
}
.calculator-container {
max-width: 800px;
margin: 30px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
border: 1px solid #e0e0e0;
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h1, h2 {
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text-align: center;
margin-bottom: 20px;
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.input-section, .result-section {
margin-bottom: 30px;
padding: 20px;
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.explanation {
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Augmented Matrix Calculator
Enter the coefficients of your system of linear equations to form an augmented matrix and see its row-echelon form.
Result: Row-Echelon Form (Reduced)
Enter dimensions and coefficients above.
Understanding Augmented Matrices and Row-Echelon Form
An augmented matrix is a matrix formed by adding the columns of coefficients from a system of linear equations and the constant terms on the right-hand side of the equations. It's a compact way to represent a system of equations, making it easier to solve using matrix operations.
For a system of equations like:
a₁₁x₁ + a₁₂x₂ + ... + a₁nxn = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂nxn = b₂
...
am₁x₁ + am₂x₂ + ... + amnxn = bm
The corresponding augmented matrix is:
[ A | B ] =
[[a₁₁, a₁₂, ..., a₁n | b₁],
[a₂₁, a₂₂, ..., a₂n | b₂],
...,
[am₁, am₂, ..., amn | bm]]
Where A is the coefficient matrix and B is the column vector of constants.
Row-Echelon Form (REF) and Reduced Row-Echelon Form (RREF)
The goal of using an augmented matrix is often to solve the system. This is achieved by transforming the matrix into its Row-Echelon Form (REF) or, more commonly, Reduced Row-Echelon Form (RREF) using elementary row operations. These operations include:
- Swapping two rows.
- Multiplying a row by a non-zero scalar.
- Adding a multiple of one row to another row.
A matrix is in RREF if it satisfies the following conditions:
- All zero rows are at the bottom of the matrix.
- The first non-zero entry (leading entry or pivot) in each non-zero row is 1.
- Each leading 1 is the only non-zero entry in its column.
- The leading 1 in any row is to the right of the leading 1 in the row above it.
How this Calculator Works
This calculator takes the dimensions of your augmented matrix (number of equations and number of coefficients + constants) and then allows you to input the individual elements. It then applies Gaussian elimination and back-substitution (or Gauss-Jordan elimination) using elementary row operations to transform the matrix into its Reduced Row-Echelon Form (RREF). The output displays this RREF matrix.
Use Cases
- Solving systems of linear equations.
- Determining if a system has a unique solution, no solution, or infinitely many solutions.
- Finding the inverse of a matrix (by augmenting with an identity matrix).
- Calculating the rank of a matrix.
- Working with linear algebra concepts in various fields like engineering, physics, computer graphics, and economics.
// Function to generate input fields for the matrix
function generateMatrixInputs() {
var numRows = parseInt(document.getElementById("rows").value);
var numCols = parseInt(document.getElementById("cols").value);
var matrixInputsDiv = document.getElementById("matrix-inputs");
matrixInputsDiv.innerHTML = "; // Clear previous inputs
if (isNaN(numRows) || isNaN(numCols) || numRows < 1 || numCols 10 || numCols > 11) {
document.getElementById("error-message").innerText = "Please enter valid dimensions (Rows: 1-10, Columns: 2-11).";
return;
}
document.getElementById("error-message").innerText = ""; // Clear error message
var table = document.createElement('table');
table.style.width = '100%';
table.style.borderCollapse = 'collapse';
for (var i = 0; i < numRows; i++) {
var row = table.insertRow();
for (var j = 0; j < numCols; j++) {
var cell = row.insertCell();
var input = document.createElement('input');
input.type = 'number';
input.step = 'any'; // Allow decimals
input.className = 'matrix-element';
input.setAttribute('data-row', i);
input.setAttribute('data-col', j);
input.placeholder = 'a' + (i + 1) + (j + 1);
input.style.width = '95%'; // Adjust width to fit cell padding
input.style.padding = '8px';
input.style.margin = '2px';
input.style.border = '1px solid #ddd';
input.style.borderRadius = '4px';
input.style.textAlign = 'center';
// Add separator line visually
if (j === numCols – 2) {
input.style.borderRight = '2px solid #004a99';
input.style.marginRight = '5px';
}
cell.appendChild(input);
}
cell.style.paddingRight = '10px'; // Add padding to separate last column
}
matrixInputsDiv.appendChild(table);
}
// Function to get the matrix from input fields
function getMatrixFromInputs() {
var numRows = parseInt(document.getElementById("rows").value);
var numCols = parseInt(document.getElementById("cols").value);
var matrix = [];
for (var i = 0; i < numRows; i++) {
matrix[i] = [];
for (var j = 0; j < numCols; j++) {
var input = document.querySelector('input[data-row="' + i + '"][data-col="' + j + '"]');
var value = parseFloat(input.value);
// Handle empty inputs as 0, but check for NaN after parseFloat
if (isNaN(value)) {
matrix[i][j] = 0;
} else {
matrix[i][j] = value;
}
}
}
return matrix;
}
// Function to display the matrix in the result area
function displayMatrix(matrix, elementId) {
var resultDiv = document.getElementById(elementId);
var numRows = matrix.length;
if (numRows === 0) {
resultDiv.innerHTML = 'Matrix is empty.';
return;
}
var numCols = matrix[0].length;
var table = document.createElement('table');
table.style.width = '100%';
table.style.borderCollapse = 'collapse';
table.style.margin = '10px auto';
for (var i = 0; i < numRows; i++) {
var row = table.insertRow();
for (var j = 0; j < numCols; j++) {
var cell = row.insertCell();
var value = matrix[i][j];
// Format numbers for better readability, especially small/large numbers
if (Math.abs(value) < 1e-9) { // Treat very small numbers as zero
value = 0;
} else {
value = parseFloat(value.toFixed(6)); // Limit decimal places
}
cell.textContent = value;
cell.style.padding = '10px';
cell.style.border = '1px solid #eee';
cell.style.textAlign = 'center';
if (j === numCols – 2) {
cell.style.borderRight = '2px solid #004a99';
}
}
}
resultDiv.innerHTML = ''; // Clear previous content
resultDiv.appendChild(table);
}
// Gauss-Jordan Elimination to find RREF
function calculateRREF() {
var errorMessageDiv = document.getElementById("error-message");
errorMessageDiv.innerText = ""; // Clear previous errors
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = 'Calculating…';
var numRows = parseInt(document.getElementById("rows").value);
var numCols = parseInt(document.getElementById("cols").value);
if (isNaN(numRows) || isNaN(numCols) || numRows < 1 || numCols 10 || numCols > 11) {
errorMessageDiv.innerText = "Invalid dimensions. Rows: 1-10, Columns: 2-11.";
resultDiv.innerHTML = 'Enter dimensions and coefficients.';
return;
}
var matrix = getMatrixFromInputs();
var epsilon = 1e-10; // Tolerance for floating point comparisons
var lead = 0; // Current pivot column
for (var r = 0; r = numCols) {
break;
}
var i = r;
// Find a row with a non-zero entry in the current pivot column
while (Math.abs(matrix[i][lead]) < epsilon) {
i++;
if (i === numRows) { // Reached end of rows for this column
i = r;
lead++;
if (lead === numCols) {
// Reached end of columns, break outer loop
lead = -1; // Sentinel value to indicate completion
break;
}
}
}
if (lead === -1) break; // Exit outer loop if all columns processed
// Swap rows if necessary to bring the pivot element to the current row
var temp = matrix[i];
matrix[i] = matrix[r];
matrix[r] = temp;
// Normalize the pivot row (make the pivot element 1)
var pivotValue = matrix[r][lead];
for (var j = 0; j < numCols; j++) {
matrix[r][j] /= pivotValue;
}
// Eliminate other entries in the pivot column
for (var i = 0; i < numRows; i++) {
if (i !== r) {
var factor = matrix[i][lead];
for (var j = 0; j < numCols; j++) {
matrix[i][j] -= factor * matrix[r][j];
}
}
}
lead++; // Move to the next column for the next pivot
}
// Clean up near-zero values to improve display
for (var r = 0; r < numRows; r++) {
for (var c = 0; c < numCols; c++) {
if (Math.abs(matrix[r][c]) < epsilon) {
matrix[r][c] = 0;
}
}
}
displayMatrix(matrix, "result");
}
// Initial generation of matrix inputs on page load
window.onload = generateMatrixInputs;