Matrix Calculator
Perform matrix operations with detailed calculations and explanations
Matrix Operations
Matrix Calculations
Matrix A
Matrix B
Understanding Matrix Operations
What is a Matrix?
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are fundamental in linear algebra and have extensive applications in mathematics, physics, computer science, and engineering.
Matrix Notation:
A = [aᵢⱼ] where i = row, j = column
m×n matrix has m rows and n columns
Example: A = [[1, 2], [3, 4]] is a 2×2 matrix
Common Matrix Operations
Matrix operations follow specific rules and have important properties that make them useful for solving systems of equations, transformations, and data analysis.
- •Addition/Subtraction: Element-wise operations for same-sized matrices
- •Multiplication: Row-by-column multiplication (not commutative)
- •Determinant: Scalar value representing matrix properties
- •Inverse: Matrix that when multiplied gives identity matrix
Matrix Operation Formulas
Addition: C = A + B
cᵢⱼ = aᵢⱼ + bᵢⱼ
Multiplication: C = A × B
cᵢⱼ = Σ(aᵢₖ × bₖⱼ) for k = 1 to n
2×2 Determinant: |A| = a₁₁a₂₂ - a₁₂a₂₁
2×2 Inverse: A⁻¹ = (1/|A|) × [[a₂₂, -a₁₂], [-a₂₁, a₁₁]]
Real-World Applications
Computer Graphics
3D transformations and animations
- • Rotation matrices
- • Scaling transformations
- • Translation matrices
- • 3D projections
Linear Systems
Solving equations simultaneously
- • Gaussian elimination
- • Cramer's rule
- • LU decomposition
- • Eigenvalue problems
Data Science
Machine learning and statistics
- • Covariance matrices
- • Principal component analysis
- • Neural networks
- • Data transformations
Physics
Quantum mechanics and relativity
- • Pauli matrices
- • Lorentz transformations
- • State vectors
- • Hamiltonian operators
Engineering
Structural analysis and design
- • Finite element analysis
- • Stress-strain matrices
- • Circuit analysis
- • Control systems
Economics
Input-output models and optimization
- • Leontief models
- • Portfolio optimization
- • Game theory
- • Markov chains
Special Types of Matrices
Common Special Matrices
Identity Matrix (I)
Diagonal elements = 1, others = 0
Properties: A × I = I × A = A
Zero Matrix (0)
All elements = 0
Properties: A + 0 = A, A × 0 = 0
Diagonal Matrix
Non-zero elements only on diagonal
Properties: Easy to compute inverse and determinant
Symmetric Matrix
A = Aᵀ (transpose equals itself)
Properties: Real eigenvalues, orthogonal eigenvectors
Advanced Properties
Orthogonal Matrix
Aᵀ × A = I (columns are orthonormal)
Used in rotations and reflections
Positive Definite Matrix
xᵀAx > 0 for all non-zero x
Important in optimization
Singular Matrix
Determinant = 0, no inverse
Indicates linear dependence
Hermitian Matrix
A = A* (conjugate transpose)
Real eigenvalues in quantum mechanics