function calculateMatrixMultiplication() {
var a11 = parseFloat(document.getElementById('a11').value);
var a12 = parseFloat(document.getElementById('a12').value);
var a21 = parseFloat(document.getElementById('a21').value);
var a22 = parseFloat(document.getElementById('a22').value);
var b11 = parseFloat(document.getElementById('b11').value);
var b12 = parseFloat(document.getElementById('b12').value);
var b21 = parseFloat(document.getElementById('b21').value);
var b22 = parseFloat(document.getElementById('b22').value);
if (isNaN(a11) || isNaN(a12) || isNaN(a21) || isNaN(a22) ||
isNaN(b11) || isNaN(b12) || isNaN(b21) || isNaN(b22)) {
document.getElementById('c11').innerText = 'C[1,1]: Invalid Input';
document.getElementById('c12').innerText = 'C[1,2]: Invalid Input';
document.getElementById('c21').innerText = 'C[2,1]: Invalid Input';
document.getElementById('c22').innerText = 'C[2,2]: Invalid Input';
return;
}
// Calculate elements of the resulting matrix C = A * B
var c11 = (a11 * b11) + (a12 * b21);
var c12 = (a11 * b12) + (a12 * b22);
var c21 = (a21 * b11) + (a22 * b21);
var c22 = (a21 * b12) + (a22 * b22);
document.getElementById('c11').innerText = 'C[1,1]: ' + c11;
document.getElementById('c12').innerText = 'C[1,2]: ' + c12;
document.getElementById('c21').innerText = 'C[2,1]: ' + c21;
document.getElementById('c22').innerText = 'C[2,2]: ' + c22;
}
Understanding Matrix Multiplication
Matrices are fundamental mathematical objects used across various fields, including physics, engineering, computer graphics, and economics. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Operations like addition, subtraction, and multiplication can be performed on matrices, each with specific rules.
What is a Matrix?
A matrix is typically denoted by a capital letter, and its elements are enclosed in brackets. For example, a 2×2 matrix (read as "two by two") has two rows and two columns:
A = | a11 a12 |
| a21 a22 |
Here, aij refers to the element in the i-th row and j-th column. For instance, a11 is the element in the first row, first column.
The Process of Matrix Multiplication
Matrix multiplication is one of the most important operations. Unlike scalar multiplication (where every element is multiplied by a single number), matrix multiplication involves a more complex process of combining rows from the first matrix with columns from the second matrix. For two matrices A and B to be multiplied to form a product matrix C (C = A * B), the number of columns in matrix A must be equal to the number of rows in matrix B.
For two 2×2 matrices:
A = | a11 a12 | B = | b11 b12 |
| a21 a22 | | b21 b22 |
The resulting product matrix C will also be a 2×2 matrix, with its elements calculated as follows:
- C11 = (a11 * b11) + (a12 * b21)
- C12 = (a11 * b12) + (a12 * b22)
- C21 = (a21 * b11) + (a22 * b21)
- C22 = (a21 * b12) + (a22 * b22)
Each element Cij is found by taking the dot product of the i-th row of matrix A and the j-th column of matrix B.
Using the 2×2 Matrix Multiplication Calculator
Our calculator simplifies this process for 2×2 matrices. Simply input the four elements for Matrix A and the four elements for Matrix B into their respective fields. The calculator will then apply the matrix multiplication rules to determine the elements of the product matrix C.
Example Calculation:
Let's use the default values provided in the calculator:
A = | 1 2 | B = | 5 6 |
| 3 4 | | 7 8 |
Applying the multiplication rules:
- 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
Therefore, the product matrix C is:
C = | 19 22 |
| 43 50 |
This calculator is a handy tool for students, engineers, and anyone needing to quickly perform 2×2 matrix multiplication without manual calculation errors.
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