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Matrix Calculator
Perform matrix operations including addition, multiplication, determinant, and inverse
Matrix Operations
Matrix A
Matrix B
How to Use the Matrix Calculator
Getting Started
- Step 1: Select the matrix operation you want to perform
- Step 2: Choose the matrix size (2×2, 3×3, or 4×4)
- Step 3: Enter the matrix elements (numbers only)
- Step 4: For two-matrix operations, fill in both matrices
- Step 5: Click Calculate to see the result and steps
Matrix Operations
- Addition: Adds corresponding elements of two matrices
- Multiplication: Matrix product following row-column rule
- Determinant: Scalar value that characterizes the matrix
- Inverse: Matrix A⁻¹ such that A × A⁻¹ = I
- Transpose: Flips matrix over its diagonal
Key Concepts
- Square Matrix: Same number of rows and columns
- Identity Matrix: Square matrix with 1s on diagonal, 0s elsewhere
- Singular Matrix: Matrix with determinant = 0 (no inverse)
- Eigenvalues: Special scalars associated with linear transformations
Applications
- Linear Systems: Solving systems of linear equations
- Transformations: Geometric transformations in computer graphics
- Physics: Quantum mechanics and state transformations
- Economics: Input-output models and optimization
Frequently Asked Questions
When can I multiply two matrices?
Matrix multiplication A × B is possible when the number of columns in A equals the number of rows in B. The result has dimensions (rows of A) × (columns of B).
Why does my matrix have no inverse?
A matrix has no inverse when its determinant is zero (singular matrix). This means the matrix transformation is not reversible.
What is the determinant used for?
The determinant tells us if a matrix is invertible (det ≠ 0), the scaling factor of the linear transformation, and orientation preservation (positive vs negative determinant).
How accurate are the calculations?
The calculator uses double-precision floating-point arithmetic, providing accuracy to about 15 decimal places. For very large matrices or ill-conditioned problems, numerical errors may accumulate.
Related Calculators
- Algebra Solver - For solving systems of linear equations
- Vector Calculator - For vector operations and dot products
- Linear Programming Solver - For optimization problems
- Statistical Test Calculator - For statistical analysis using matrices