Matrix Operations Calculator (2×2)
This calculator allows you to perform basic operations on 2×2 matrices, including addition, subtraction, multiplication, determinant, and transpose.
Matrix A
Matrix B
Result:
" + operationName + " Result:
"; output += ""; output += "[ " + matrix[0][0].toFixed(4).padStart(8) + " " + matrix[0][1].toFixed(4).padStart(8) + " ]\n"; output += "[ " + matrix[1][0].toFixed(4).padStart(8) + " " + matrix[1][1].toFixed(4).padStart(8) + " ]"; output += ""; resultDiv.innerHTML = output; } function displayScalarResult(scalar, operationName) { var resultDiv = document.getElementById('matrixResultDisplay'); if (isNaN(scalar)) { resultDiv.innerHTML = "Error: Invalid input for " + operationName + ". Please enter valid numbers for all matrix elements."; return; } resultDiv.innerHTML = "" + operationName + " Result:
" + scalar.toFixed(4) + ""; } function calculateMatrixAddition() { var A = getMatrixA(); var B = getMatrixB(); if (!A || !B) { displayMatrixResult(null, "Matrix Addition"); return; } var C = [ [A[0][0] + B[0][0], A[0][1] + B[0][1]], [A[1][0] + B[1][0], A[1][1] + B[1][1]] ]; displayMatrixResult(C, "Matrix Addition (A + B)"); } function calculateMatrixSubtraction() { var A = getMatrixA(); var B = getMatrixB(); if (!A || !B) { displayMatrixResult(null, "Matrix Subtraction"); return; } var C = [ [A[0][0] – B[0][0], A[0][1] – B[0][1]], [A[1][0] – B[1][0], A[1][1] – B[1][1]] ]; displayMatrixResult(C, "Matrix Subtraction (A – B)"); } function calculateMatrixMultiplication() { var A = getMatrixA(); var B = getMatrixB(); if (!A || !B) { displayMatrixResult(null, "Matrix Multiplication"); return; } var C = [ [A[0][0] * B[0][0] + A[0][1] * B[1][0], A[0][0] * B[0][1] + A[0][1] * B[1][1]], [A[1][0] * B[0][0] + A[1][1] * B[1][0], A[1][0] * B[0][1] + A[1][1] * B[1][1]] ]; displayMatrixResult(C, "Matrix Multiplication (A * B)"); } function calculateMatrixDeterminant() { var A = getMatrixA(); if (!A) { displayScalarResult(NaN, "Determinant of A"); return; } var det = A[0][0] * A[1][1] – A[0][1] * A[1][0]; displayScalarResult(det, "Determinant of A"); } function calculateMatrixTranspose() { var A = getMatrixA(); if (!A) { displayMatrixResult(null, "Transpose of A"); return; } var A_T = [ [A[0][0], A[1][0]], [A[0][1], A[1][1]] ]; displayMatrixResult(A_T, "Transpose of A"); }Understanding Matrix Operations
Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are fundamental tools in mathematics, physics, engineering, computer graphics, and many other fields for representing linear transformations, systems of linear equations, and data.
What is a Matrix?
A matrix is typically denoted by a capital letter (e.g., A, B) and its elements by lowercase letters with subscripts indicating their position (e.g., aij for the element in the i-th row and j-th column). A 2×2 matrix, like the one used in this calculator, has two rows and two columns:
A = | a11 a12 |
| a21 a22 |
Basic Matrix Operations Explained
1. Matrix Addition (A + B)
To add two matrices, they must have the same dimensions. You simply add the corresponding elements. For two 2×2 matrices A and B, their sum C = A + B is:
C = | a11+b11 a12+b12 |
| a21+b21 a22+b22 |
Example: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].
2. Matrix Subtraction (A – B)
Similar to addition, matrices must have the same dimensions. You subtract the corresponding elements. For two 2×2 matrices A and B, their difference C = A – B is:
C = | a11-b11 a12-b12 |
| a21-b21 a22-b22 |
Example: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A – B = [[1-5, 2-6], [3-7, 4-8]] = [[-4, -4], [-4, -4]].
3. Matrix Multiplication (A * B)
Matrix multiplication is more complex. For two matrices A (m x n) and B (n x p) to be multiplied, the number of columns in A must equal the number of rows in B. The resulting matrix C will have dimensions (m x p). For two 2×2 matrices A and B, their product C = A * B is:
C = | a11b11+a12b21 a11b12+a12b22 |
| a21b11+a22b21 a21b12+a22b22 |
Each element cij is the dot product of the i-th row of A and the j-th column of B.
Example: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A * B = [[(1*5)+(2*7), (1*6)+(2*8)], [(3*5)+(4*7), (3*6)+(4*8)]] = [[5+14, 6+16], [15+28, 18+32]] = [[19, 22], [43, 50]].
4. Determinant of a Matrix (det A)
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. For a 2×2 matrix A:
det A = a11a22 - a12a21
Example: If A = [[1, 2], [3, 4]], then det A = (1*4) – (2*3) = 4 – 6 = -2.
5. Transpose of a Matrix (AT)
The transpose of a matrix is obtained by flipping the matrix over its diagonal; that is, it switches the row and column indices of the matrix. For a 2×2 matrix A, its transpose AT is:
AT = | a11 a21 |
| a12 a22 |
Example: If A = [[1, 2], [3, 4]], then AT = [[1, 3], [2, 4]].
Applications of Matrix Operations
- Computer Graphics: Used for transformations like rotation, scaling, and translation of objects in 2D and 3D space.
- Solving Systems of Linear Equations: Matrices provide a concise way to represent and solve systems of equations.
- Physics and Engineering: Describing forces, stresses, strains, and quantum mechanics.
- Data Science and Machine Learning: Fundamental for algorithms like linear regression, principal component analysis (PCA), and neural networks.
- Cryptography: Used in encoding and decoding messages.
This calculator provides a quick way to verify your manual calculations for 2×2 matrices, helping you understand these fundamental operations better.