Matrix Multiplication Calculator
Matrix A
Matrix B
Result Matrix (A x B)
Result Matrix (A x B)
"; // Clear previous result // Validate dimensions if (isNaN(matrixARows) || matrixARows < 1 || isNaN(matrixACols) || matrixACols < 1 || isNaN(matrixBRows) || matrixBRows < 1 || isNaN(matrixBCols) || matrixBCols < 1) { errorMessage.innerHTML = "Please ensure all matrix dimensions are valid positive numbers."; return; } // Matrix multiplication condition: A's columns must equal B's rows if (matrixACols !== matrixBRows) { errorMessage.innerHTML = "Error: For matrix multiplication (A x B), the number of columns in Matrix A (" + matrixACols + ") must equal the number of rows in Matrix B (" + matrixBRows + "). Please adjust dimensions."; return; } // Read Matrix A values var matrixA = []; for (var i = 0; i < matrixARows; i++) { matrixA[i] = []; for (var j = 0; j < matrixACols; j++) { var inputId = "matrixA_" + i + "_" + j; var inputElement = document.getElementById(inputId); if (!inputElement) { errorMessage.innerHTML = "Matrix A inputs not generated. Please click 'Generate Matrix A Inputs'."; return; } var value = parseFloat(inputElement.value); if (isNaN(value)) { errorMessage.innerHTML = "Please enter valid numbers for all elements in Matrix A. Found non-numeric value at A[" + i + "][" + j + "]."; return; } matrixA[i][j] = value; } } // Read Matrix B values var matrixB = []; for (var i = 0; i < matrixBRows; i++) { matrixB[i] = []; for (var j = 0; j < matrixBCols; j++) { var inputId = "matrixB_" + i + "_" + j; var inputElement = document.getElementById(inputId); if (!inputElement) { errorMessage.innerHTML = "Matrix B inputs not generated. Please click 'Generate Matrix B Inputs'."; return; } var value = parseFloat(inputElement.value); if (isNaN(value)) { errorMessage.innerHTML = "Please enter valid numbers for all elements in Matrix B. Found non-numeric value at B[" + i + "][" + j + "]."; return; } matrixB[i][j] = value; } } // Perform Matrix Multiplication var matrixC = []; for (var i = 0; i < matrixARows; i++) { matrixC[i] = []; for (var j = 0; j < matrixBCols; j++) { var sum = 0; for (var k = 0; k < matrixACols; k++) { // matrixACols is equal to matrixBRows sum += matrixA[i][k] * matrixB[k][j]; } matrixC[i][j] = sum; } } // Display Result var resultTable = "| " + matrixC[i][j].toFixed(4) + " | "; // Format to 4 decimal places for precision } resultTable += "
Understanding Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a new matrix. Unlike scalar multiplication (where each element is multiplied by a single number) or element-wise matrix multiplication (Hadamard product), matrix multiplication involves a more complex process of dot products between rows and columns.
Conditions for Matrix Multiplication
For two matrices, Matrix A and Matrix B, to be multiplied (A x B), a specific condition must be met:
- The number of columns in the first matrix (Matrix A) must be equal to the number of rows in the second matrix (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 for the product A x B to be defined, n must be equal to p. The resulting product matrix, let's call it Matrix C, will then have dimensions m x q.
How Matrix Multiplication Works (The Formula)
Each element Cij in the product 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. This means you multiply corresponding elements from the row and column and then sum those products.
Mathematically, if A = [aik] and B = [bkj], then the element Cij of the product matrix C = A x B is given by:
Cij = Σ (aik * bkj)where the summation is performed over k from 1 to n (or p, since n = p).
Step-by-Step Example
Let's illustrate with a concrete example:
Matrix A (2×2):
[ 1 2 ]
[ 3 4 ]Matrix B (2×2):
[ 5 6 ]
[ 7 8 ]Here, Matrix A has 2 columns and Matrix B has 2 rows, so multiplication is possible. The resulting matrix C will be 2×2.
Calculating C11 (first row, first column):
Take the first row of A [1 2] and the first column of B [5 7].
C11 = (1 * 5) + (2 * 7) = 5 + 14 = 19Calculating C12 (first row, second column):
Take the first row of A [1 2] and the second column of B [6 8].
C12 = (1 * 6) + (2 * 8) = 6 + 16 = 22Calculating C21 (second row, first column):
Take the second row of A [3 4] and the first column of B [5 7].
C21 = (3 * 5) + (4 * 7) = 15 + 28 = 43Calculating C22 (second row, second column):
Take the second row of A [3 4] and the second column of B [6 8].
C22 = (3 * 6) + (4 * 8) = 18 + 32 = 50Resulting Matrix C (A x B):
[ 19 22 ]
[ 43 50 ]Applications of Matrix Multiplication
Matrix multiplication is not just a theoretical concept; it has wide-ranging practical applications across various fields:
- Computer Graphics: Used extensively for 2D and 3D transformations such as rotation, scaling, translation, and projection of objects.
- Solving Systems of Linear Equations: Matrix multiplication provides a compact way to represent and solve systems of linear equations, which are common in engineering, physics, and economics.
- Physics and Engineering: Essential in quantum mechanics, optics, structural analysis, and electrical circuit analysis.
- Data Science and Machine Learning: Forms the backbone of many algorithms, including neural networks, principal component analysis (PCA), and various data transformations.
- Economics: Used in input-output models to analyze interdependencies between different sectors of an economy.
This calculator allows you to quickly perform matrix multiplication for matrices of various sizes, helping you understand and verify the results of this powerful mathematical operation.