Matrix Calculator
Perform matrix operations like addition, multiplication, determinant, and inverse with our online matrix calculator.
Matrix Operations
Configure matrices and select operation
2×2
2×2
Matrix Operations:
• Addition/Subtraction: Same dimensions required
• Multiplication: A cols = B rows
• Determinant/Inverse: Square matrices only
Understanding Matrix Operations
Matrices are rectangular arrays of numbers that are fundamental in linear algebra, computer graphics, engineering, and data science. Understanding matrix operations is essential for solving systems of linear equations, transformations, and many computational problems.
Basic Matrix Operations
- Addition/Subtraction: Element-wise operations requiring same dimensions
- Multiplication: Row-by-column multiplication with specific dimension requirements
- Transpose: Flipping rows and columns of a matrix
- Determinant: Scalar value that provides important matrix properties
- Inverse: Matrix that when multiplied gives the identity matrix
Matrix Properties
- Square matrices have equal rows and columns
- Identity matrix has 1s on diagonal, 0s elsewhere
- Singular matrices have determinant = 0 and no inverse
- Orthogonal matrices preserve lengths and angles
Applications of Matrices
Computer Graphics
- 3D transformations and rotations
- Scaling and translation operations
- Projection matrices for rendering
- Animation and interpolation
Engineering & Science
- Solving systems of linear equations
- Circuit analysis and network theory
- Structural analysis and mechanics
- Quantum mechanics and physics
Matrix Operation Rules
| Operation | Requirements | Result Dimensions | Properties |
|---|---|---|---|
| Addition (A + B) | Same dimensions | Same as inputs | Commutative, Associative |
| Subtraction (A - B) | Same dimensions | Same as inputs | Not commutative |
| Multiplication (A × B) | A cols = B rows | A rows × B cols | Not commutative, Associative |
| Transpose (A^T) | Any matrix | Flipped dimensions | (A^T)^T = A |
| Determinant (det(A)) | Square matrix | Scalar value | det(AB) = det(A)det(B) |
| Inverse (A^-1) | Square, det ≠ 0 | Same as input | AA^-1 = I |