Matrix Calculator with Steps
Matrix Input
x
Addition (+)
Subtraction (-)
Multiplication (*)
Transpose (AT)
Determinant (det(A))
Inverse (A-1)
x
Result
Note:
- Matrix dimensions are limited to 10×10 for performance and usability.
- For operations like multiplication, dimensions must be compatible (A: m x n, B: n x p).
- Determinant and Inverse are only defined for square matrices (n x n).
- This calculator provides step-by-step explanations for common matrix operations.
Understanding Matrix Operations
Matrices are fundamental tools in various fields, including mathematics, physics, computer science, engineering, and economics. They are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Matrix operations allow us to manipulate and analyze data represented in this format, leading to powerful insights and solutions to complex problems.
Matrix Dimensions
A matrix is defined by its dimensions, denoted as "rows x columns". For example, a matrix with 3 rows and 2 columns is a 3×2 matrix.
Common Matrix Operations
1. Matrix Addition and Subtraction
To add or subtract two matrices, they must have the exact same dimensions. The operation is performed element-wise: the element in the i-th row and j-th column of the resulting matrix is the sum or difference of the corresponding elements in the original matrices.
Formula: If C = A + B, then \(C_{ij} = A_{ij} + B_{ij}\). If C = A – B, then \(C_{ij} = A_{ij} – B_{ij}\).
Example:
Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].
A – B = [[1-5, 2-6], [3-7, 4-8]] = [[-4, -4], [-4, -4]].
2. Matrix Multiplication
Matrix multiplication is more complex. To multiply matrix A (m x n) by matrix B (p x q), the number of columns in A (n) must equal the number of rows in B (p). The resulting matrix C will have dimensions m x q.
Formula: If C = A * B, then \(C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}\). This means each element \(C_{ij}\) is calculated by taking the dot product of the i-th row of A and the j-th column of B.
Example:
Let A = [[1, 2], [3, 4]] (2×2) and B = [[5, 6], [7, 8]] (2×2).
C11 = (1 * 5) + (2 * 7) = 5 + 14 = 19
C12 = (1 * 6) + (2 * 8) = 6 + 16 = 22
C21 = (3 * 5) + (4 * 7) = 15 + 28 = 43
C22 = (3 * 6) + (4 * 8) = 18 + 32 = 50
C = [[19, 22], [43, 50]].
3. Matrix Transpose
The transpose of a matrix A, denoted as AT, is obtained by interchanging its rows and columns. If A is an m x n matrix, its transpose AT is an n x m matrix.
Formula: \(A^T_{ij} = A_{ji}\).
Example:
If A = [[1, 2, 3], [4, 5, 6]] (2×3), then
AT = [[1, 4], [2, 5], [3, 6]] (3×2).
4. Determinant
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible. The calculation is complex for larger matrices but follows specific rules.
For a 2×2 matrix: If A = [[a, b], [c, d]], then det(A) = ad – bc.
Example:
If A = [[4, 7], [2, 6]], then det(A) = (4 * 6) – (7 * 2) = 24 – 14 = 10.
For larger matrices (3×3 and above), methods like cofactor expansion or row reduction are used, which are computationally intensive.
5. Matrix Inverse
The inverse of a square matrix A, denoted as A-1, is a matrix such that when multiplied by A, it yields the identity matrix (I). An inverse exists only if the determinant of the matrix is non-zero.
Formula (for 2×2): If A = [[a, b], [c, d]], then \(A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\).
Example:
If A = [[4, 7], [2, 6]], det(A) = 10.
\(A^{-1} = \frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{bmatrix}\).
Calculating the inverse for matrices larger than 2×2 typically involves methods like Gauss-Jordan elimination or adjugate matrices.
Use Cases
- Solving Systems of Linear Equations: Matrices are extensively used to represent and solve systems of linear equations, common in scientific modeling and optimization problems.
- Computer Graphics: Transformations like translation, rotation, and scaling of objects in 2D and 3D graphics are performed using matrix multiplication.
- Data Analysis and Machine Learning: In fields like machine learning, matrices represent datasets, feature vectors, and model parameters. Operations like PCA (Principal Component Analysis) and regression rely heavily on matrix computations.
- Physics and Engineering: Used in structural analysis, circuit analysis, quantum mechanics, and signal processing.
';
for (var i = 0; i < rows; i++) {
for (var j = 0; j < cols; j++) {
matrixHtml += '';
}
}
matrixHtml += '
';
container.innerHTML = matrixHtml;
}
// Helper function to format matrix for display
function formatMatrix(matrix) {
if (!matrix) return "N/A";
return matrix.map(row => '[' + row.map(val => val.toFixed(4)).join(', ') + ']').join('\n');
}
// Function to generate the calculation steps string
function generateSteps(operation, matrixA, matrixB, result, intermediateSteps = []) {
var steps = `Operation: ${operation}\n`;
steps += `Matrix A:\n${formatMatrix(matrixA)}\n`;
if (matrixB) steps += `Matrix B:\n${formatMatrix(matrixB)}\n`;
if (intermediateSteps.length > 0) {
steps += "\nIntermediate Steps:\n";
intermediateSteps.forEach((step, index) => {
steps += `${index + 1}. ${step}\n`;
});
}
steps += `\nFinal Result:\n${formatMatrix(result)}\n`;
return steps;
}
// Initialize matrix inputs on page load or dimension change
function initializeMatrixInputs() {
var rowsA = parseInt(document.getElementById('rowsA').value);
var colsA = parseInt(document.getElementById('colsA').value);
createMatrixInputs(rowsA, colsA, 'matrixAContainer');
var operation = document.getElementById('operation').value;
if (operation === 'add' || operation === 'subtract' || operation === 'multiply') {
document.getElementById('matrixBSection').style.display = 'flex';
var rowsB = parseInt(document.getElementById('rowsB').value);
var colsB = parseInt(document.getElementById('colsB').value);
createMatrixInputs(rowsB, colsB, 'matrixBContainer');
} else {
document.getElementById('matrixBSection').style.display = 'none';
}
}
// Event listeners for dimension changes to update input fields
document.getElementById('rowsA').addEventListener('change', function() {
var rowsA = parseInt(this.value);
var colsA = parseInt(document.getElementById('colsA').value);
createMatrixInputs(rowsA, colsA, 'matrixAContainer');
updateMatrixBVisibility();
});
document.getElementById('colsA').addEventListener('change', function() {
var rowsA = parseInt(document.getElementById('rowsA').value);
var colsA = parseInt(this.value);
createMatrixInputs(rowsA, colsA, 'matrixAContainer');
updateMatrixBVisibility();
});
document.getElementById('rowsB').addEventListener('change', function() {
var rowsB = parseInt(this.value);
var colsB = parseInt(document.getElementById('colsB').value);
createMatrixInputs(rowsB, colsB, 'matrixBContainer');
});
document.getElementById('colsB').addEventListener('change', function() {
var rowsB = parseInt(document.getElementById('rowsB').value);
var colsB = parseInt(this.value);
createMatrixInputs(rowsB, colsB, 'matrixBContainer');
});
document.getElementById('operation').addEventListener('change', updateMatrixBVisibility);
function updateMatrixBVisibility() {
var operation = document.getElementById('operation').value;
if (operation === 'add' || operation === 'subtract' || operation === 'multiply') {
document.getElementById('matrixBSection').style.display = 'flex';
var rowsA = parseInt(document.getElementById('rowsA').value);
var colsA = parseInt(document.getElementById('colsA').value);
var rowsB = parseInt(document.getElementById('rowsB').value);
var colsB = parseInt(document.getElementById('colsB').value);
if (operation === 'multiply') {
document.getElementById('rowsB').value = colsA; // Auto-adjust B rows for multiplication
document.getElementById('rowsB').disabled = true; // Make it read-only
rowsB = colsA; // Update local var
} else {
document.getElementById('rowsB').disabled = false;
}
createMatrixInputs(rowsB, colsB, 'matrixBContainer');
} else {
document.getElementById('matrixBSection').style.display = 'none';
}
}
// — Calculation Logic —
function calculateMatrix() {
document.getElementById('error').innerText = ";
document.getElementById('result').innerText = ";
document.getElementById('steps').innerText = ";
document.getElementById('resultSection').style.display = 'none';
var operation = document.getElementById('operation').value;
var rowsA, colsA, matrixA, matrixB;
try {
rowsA = parseInt(document.getElementById('rowsA').value);
colsA = parseInt(document.getElementById('colsA').value);
matrixA = parseMatrix(rowsA, colsA, 'matrixAContainer');
var intermediateSteps = [];
var resultMatrix = null;
if (operation === 'add' || operation === 'subtract' || operation === 'multiply') {
var rowsB = parseInt(document.getElementById('rowsB').value);
var colsB = parseInt(document.getElementById('colsB').value);
matrixB = parseMatrix(rowsB, colsB, 'matrixBContainer');
// Dimension checks for binary operations
if (rowsA !== rowsB || colsA !== colsB) {
if (operation === 'multiply' && colsA !== rowsB) {
throw new Error("For matrix multiplication, the number of columns in Matrix A must equal the number of rows in Matrix B.");
} else if (operation !== 'multiply' && (rowsA !== rowsB || colsA !== colsB)) {
throw new Error("For matrix addition and subtraction, both matrices must have the same dimensions.");
}
}
// Adjust rowsB if it was auto-set for multiplication
if (operation === 'multiply' && document.getElementById('rowsB').disabled) {
rowsB = colsA;
}
if (operation === 'add') {
resultMatrix = [];
for (var i = 0; i < rowsA; i++) {
resultMatrix[i] = [];
for (var j = 0; j ${resultMatrix[i][j]}`);
}
}
} else if (operation === 'subtract') {
resultMatrix = [];
for (var i = 0; i < rowsA; i++) {
resultMatrix[i] = [];
for (var j = 0; j ${resultMatrix[i][j]}`);
}
}
} else if (operation === 'multiply') {
resultMatrix = [];
for (var i = 0; i < rowsA; i++) {
resultMatrix[i] = [];
for (var j = 0; j < colsB; j++) {
var sum = 0;
var stepDetail = `Calculating C[${i}][${j}]: `;
for (var k = 0; k < colsA; k++) { // colsA is same as rowsB
sum += matrixA[i][k] * matrixB[k][j];
stepDetail += `(A[${i}][${k}] * B[${k}][${j}]) = (${matrixA[i][k]} * ${matrixB[k][j]})`;
if (k < colsA – 1) stepDetail += ' + ';
}
resultMatrix[i][j] = sum;
intermediateSteps.push(stepDetail + ` = ${sum}`);
}
}
}
} else if (operation === 'transpose') {
resultMatrix = [];
for (var j = 0; j < colsA; j++) {
resultMatrix[j] = [];
for (var i = 0; i 2×2.
resultMatrix = calculateDeterminantGeneral(matrixA);
intermediateSteps.push(`Determinant calculation for ${rowsA}x${rowsA} matrix (steps omitted for brevity).`);
}
} else if (operation === 'inverse') {
if (rowsA !== colsA) {
throw new Error("Inverse can only be calculated for square matrices.");
}
var det = calculateDeterminantGeneral(matrixA); // Use general determinant calculation
if (Math.abs(det) […row, …Array(rowsA).fill(0).map((_, j) => i === j ? 1 : 0)]);
var n = rowsA;
var augmentedN = 2 * n;
intermediateSteps.push(`Applying Gauss-Jordan elimination to augmented matrix [A | I].`);
for (var i = 0; i < n; i++) {
// Find pivot
var pivotRow = i;
for (var k = i + 1; k Math.abs(augmentedMatrix[pivotRow][i])) {
pivotRow = k;
}
}
// Swap rows
var temp = augmentedMatrix[i];
augmentedMatrix[i] = augmentedMatrix[pivotRow];
augmentedMatrix[pivotRow] = temp;
if (Math.abs(augmentedMatrix[i][i]) < 1e-10) {
throw new Error("Matrix is singular (pivot is zero), inverse does not exist.");
}
// Normalize pivot row
var pivotElement = augmentedMatrix[i][i];
intermediateSteps.push(`Scaling row ${i} by 1/${pivotElement} to make pivot A[${i}][${i}] = 1.`);
for (var j = i; j < augmentedN; j++) {
augmentedMatrix[i][j] /= pivotElement;
}
// Eliminate other rows
for (var k = 0; k < n; k++) {
if (k !== i) {
var factor = augmentedMatrix[k][i];
intermediateSteps.push(`Subtracting ${factor} * row ${i} from row ${k} to make A[${k}][${i}] = 0.`);
for (var j = i; j row.slice(n));
intermediateSteps.push("Gauss-Jordan elimination complete. The right half of the augmented matrix is the inverse.");
}
}
if (resultMatrix) {
document.getElementById('result').innerText = formatMatrix(resultMatrix);
document.getElementById('steps').innerText = generateSteps(operation, matrixA, matrixB, resultMatrix, intermediateSteps);
document.getElementById('resultSection').style.display = 'block';
}
} catch (e) {
document.getElementById('error').innerText = 'Error: ' + e.message;
document.getElementById('resultSection').style.display = 'block';
}
}
// General determinant calculation (using cofactor expansion) – recursive
function calculateDeterminantGeneral(matrix) {
var n = matrix.length;
if (n === 1) {
return matrix[0][0];
}
if (n === 2) {
return matrix[0][0] * matrix[1][1] – matrix[0][1] * matrix[1][0];
}
var det = 0;
for (var j = 0; j < n; j++) {
var subMatrix = [];
for (var i = 1; i colIndex !== j));
}
var sign = (j % 2 === 0) ? 1 : -1;
det += sign * matrix[0][j] * calculateDeterminantGeneral(subMatrix);
}
return det;
}
// Initial setup when the page loads
document.addEventListener('DOMContentLoaded', function() {
initializeMatrixInputs();
// Add event listeners for dimension inputs after initial setup
document.getElementById('rowsA').addEventListener('change', initializeMatrixInputs);
document.getElementById('colsA').addEventListener('change', initializeMatrixInputs);
document.getElementById('rowsB').addEventListener('change', initializeMatrixInputs);
document.getElementById('colsB').addEventListener('change', initializeMatrixInputs);
document.getElementById('operation').addEventListener('change', initializeMatrixInputs);
});