Matrix Multiplier Calculator
Input Matrices
x
x
Result Matrix (A x B)
Enter matrix dimensions and values to see the result.
Understanding Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra. It's used extensively in fields like computer graphics, physics, engineering, data science, and machine learning. The process of multiplying two matrices, say Matrix A and Matrix B, results in a new matrix, often denoted as C.
Compatibility Rule
For the multiplication A x B to be possible, the number of columns in Matrix A must equal the number of rows in Matrix B. If Matrix A has dimensions m x n (m rows, n columns) and Matrix B has dimensions p x q (p rows, q columns), then multiplication is only valid if n = p. The resulting matrix C will have dimensions m x q.
The Calculation Process
Each element in the resulting matrix C, denoted as cij (the element in the i-th row and j-th column of C), is calculated by taking the dot product of the i-th row of Matrix A and the j-th column of Matrix B. The dot product is the sum of the products of corresponding elements.
Mathematically, for a resulting matrix C = A x B:
cij = Σ (aik * bkj) for k from 1 to n (where n is the number of columns in A / rows in B)
Let's break this down:
- To find the element in the 1st row, 1st column of C (c11): Multiply each element in the 1st row of A by the corresponding element in the 1st column of B, and then sum up these products.
- To find the element in the 1st row, 2nd column of C (c12): Multiply each element in the 1st row of A by the corresponding element in the 2nd column of B, and sum up these products.
- This process continues for all rows of A and all columns of B.
Example Calculation
Let's multiply Matrix A (2×3) by Matrix B (3×2):
A = | 1 2 3 |
| 4 5 6 |
B = | 7 8 |
| 9 10 |
|11 12 |
The resulting matrix C will have dimensions 2×2.
c11 = (1 * 7) + (2 * 9) + (3 * 11) = 7 + 18 + 33 = 58
c12 = (1 * 8) + (2 * 10) + (3 * 12) = 8 + 20 + 36 = 64
c21 = (4 * 7) + (5 * 9) + (6 * 11) = 28 + 45 + 66 = 139
c22 = (4 * 8) + (5 * 10) + (6 * 12) = 32 + 50 + 72 = 154
So, the resulting matrix C is:
C = | 58 64 |
|139 154 |
Use Cases
- Computer Graphics: Transforming 3D models (rotation, scaling, translation) involves multiplying transformation matrices with vertex matrices.
- Machine Learning: Neural networks rely heavily on matrix multiplication for processing layers of data.
- Image Processing: Applying filters or transformations to images.
- Solving Systems of Linear Equations: Representing and solving linear equations.
- Quantum Mechanics: Describing state transitions and operators.
[";
for (var i = 0; i < matrix.length; i++) {
output += "[" + matrix[i].join(", ") + "]";
if (i < matrix.length - 1) {
output += ", ";
}
}
output += "]";
return output;
}
function calculateMatrixMultiplication() {
var resultDiv = document.getElementById("resultMatrix");
var errorDiv = document.getElementById("errorMessage");
errorDiv.textContent = "";
resultDiv.innerHTML = "Calculating…";
try {
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 (isNaN(rowsA) || isNaN(colsA) || rowsA < 1 || colsA < 1 ||
isNaN(rowsB) || isNaN(colsB) || rowsB < 1 || colsB < 1) {
throw new Error("Please enter valid positive integer dimensions for both matrices.");
}
var matrixAValuesInput = document.getElementById("matrixAValues").value;
var matrixBValuesInput = document.getElementById("matrixBValues").value;
if (!matrixAValuesInput || !matrixBValuesInput) {
throw new Error("Please enter values for both matrices.");
}
var matrixA = parseMatrixInput(matrixAValuesInput, rowsA, colsA);
var matrixB = parseMatrixInput(matrixBValuesInput, rowsB, colsB);
var result = multiplyMatrices(matrixA, matrixB);
resultDiv.innerHTML = formatMatrixOutput(result);
} catch (e) {
errorDiv.textContent = "Error: " + e.message;
resultDiv.innerHTML = "Calculation failed. Please check inputs.";
}
}