Matrix Multiplication Calculator
Matrix A Dimensions
Matrix B Dimensions
Resulting Matrix (A x B)
Understanding Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra used extensively in fields like computer graphics, physics, engineering, economics, and data science. It's a binary operation that produces a new matrix from two matrices, provided certain dimensional compatibility conditions are met.
The Rule of Matrix Multiplication
For two matrices, A and B, to be multiplied in the order A x B, the number of columns in matrix A must be equal to the number of rows in matrix B.
- If Matrix A has dimensions \( m \times n \) (m rows, n columns),
- And Matrix B has dimensions \( p \times q \) (p rows, q columns),
- Then the multiplication A x B is only possible if \( n = p \).
- The resulting matrix, let's call it C, will have dimensions \( m \times q \).
How to Calculate the Elements of the Resulting Matrix
Each element \( C_{ij} \) (the element in the i-th row and j-th column of the resulting matrix C) is calculated by taking the dot product of the i-th row of matrix A and the j-th column of matrix B.
Mathematically, this is represented as:
$$ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} $$
In simpler terms:
- Take the first row of matrix A.
- Take the first column of matrix B.
- Multiply the corresponding elements from the row and column.
- Sum up these products. This sum is the element in the first row, first column of the result matrix C.
- Repeat this process for the first row of A and the second column of B to get \( C_{12} \), and so on for all columns of B.
- Then, move to the second row of A and repeat the process for all columns of B to get the second row of C.
- Continue this until all rows of A have been processed against all columns of B.
Example Calculation
Let's multiply Matrix A by Matrix B:
Matrix A (2×3):
$$ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} $$
Matrix B (3×2):
$$ B = \begin{pmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{pmatrix} $$
Here, the number of columns in A (3) equals the number of rows in B (3), so multiplication is possible. The resulting matrix C will be 2×2.
Calculating the elements:
- \( C_{11} = (1 \times 7) + (2 \times 9) + (3 \times 11) = 7 + 18 + 33 = 58 \)
- \( C_{12} = (1 \times 8) + (2 \times 10) + (3 \times 12) = 8 + 20 + 36 = 64 \)
- \( C_{21} = (4 \times 7) + (5 \times 9) + (6 \times 11) = 28 + 45 + 66 = 139 \)
- \( C_{22} = (4 \times 8) + (5 \times 10) + (6 \times 12) = 32 + 50 + 72 = 154 \)
Resulting Matrix C (2×2):
$$ C = \begin{pmatrix} 58 & 64 \\ 139 & 154 \end{pmatrix} $$
Use Cases
- Computer Graphics: Transformations like rotation, scaling, and translation are often represented by matrices. Multiplying transformation matrices allows for complex sequential transformations.
- Machine Learning: Neural networks heavily rely on matrix multiplications to process data and compute outputs.
- Solving Systems of Linear Equations: Matrix multiplication is a key component in methods used to solve systems of linear equations.
- Image Processing: Applying filters and transformations to images involves matrix operations.
- Quantum Mechanics: Operations and state changes are often described using matrix multiplication.
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for (var i = 0; i < rows; i++) {
html += '
';
container.innerHTML = html;
}
function formatMatrixOutput(matrix) {
if (!matrix) return "";
var output = "";
for (var i = 0; i < matrix.length; i++) {
output += "[ ";
for (var j = 0; j < matrix[i].length; j++) {
output += matrix[i][j].toFixed(4); // Format to 4 decimal places
if (j < matrix[i].length – 1) {
output += ", ";
}
}
output += " ]\n";
}
return output.trim();
}
function calculateMatrixMultiplication() {
document.getElementById('errorMessage').textContent = '';
document.getElementById('result').textContent = '';
document.getElementById('resultSection').style.display = 'none';
var rowsA = getInputValue('rowsA');
var colsA = getInputValue('colsA');
var rowsB = getInputValue('rowsB');
var colsB = getInputValue('colsB');
if (isNaN(rowsA) || isNaN(colsA) || isNaN(rowsB) || isNaN(colsB) || rowsA <= 0 || colsA <= 0 || rowsB <= 0 || colsB <= 0) {
document.getElementById('errorMessage').textContent = 'Invalid dimensions. Please enter positive integers for rows and columns.';
return;
}
if (colsA !== rowsB) {
document.getElementById('errorMessage').textContent = 'Matrix multiplication is not possible: The number of columns in Matrix A must equal the number of rows in Matrix B.';
return;
}
var matrixA = parseMatrix('A', rowsA, colsA);
var matrixB = parseMatrix('B', rowsB, colsB);
if (matrixA === null || matrixB === null) {
document.getElementById('errorMessage').textContent = 'Invalid input in matrix elements. Please ensure all values are numbers.';
return;
}
var resultMatrix = [];
var resultRows = rowsA;
var resultCols = colsB;
for (var i = 0; i < resultRows; i++) {
resultMatrix[i] = [];
for (var j = 0; j < resultCols; j++) {
var sum = 0;
for (var k = 0; k < colsA; k++) { // colsA is same as rowsB
sum += matrixA[i][k] * matrixB[k][j];
}
resultMatrix[i][j] = sum;
}
}
document.getElementById('result').textContent = formatMatrixOutput(resultMatrix);
document.getElementById('resultSection').style.display = 'block';
}
// Initial generation of matrix inputs on page load
window.onload = function() {
generateMatrixInputs('A');
generateMatrixInputs('B');
};
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for (var j = 0; j < cols; j++) {
html += '';
}
html += '
';
}
html += '