Matrix Calculator
Perform advanced matrix operations including addition, multiplication, determinant, inverse, eigenvalues, eigenvectors, rank, and more linear algebra calculations.
Matrix Operation
Matrix A
Matrix B
Calculation Results
Select an operation and enter matrix values to see results here.
Quick Examples
Matrix Properties
What is a Matrix Calculator?
A matrix calculator is an advanced mathematical tool designed to perform operations on matrices - rectangular arrays of numbers arranged in rows and columns. Matrices are fundamental to linear algebra and are extensively used in computer graphics, engineering, physics, economics, statistics, and machine learning.
This matrix calculator supports all essential matrix operations including addition, subtraction, multiplication, determinant calculation, matrix inversion, eigenvalue computation, rank determination, and more. It provides a visual interface similar to mathematical software but accessible directly in your browser.
How to Use This Matrix Calculator
This calculator simplifies complex matrix calculations through an intuitive interface. Follow these steps to perform matrix operations:
- Select Operation: Choose from 12 different matrix operations using the operation buttons (A+B, A×B, det(A), A⁻¹, etc.).
- Set Matrix Size: Adjust the dimensions of Matrix A and Matrix B using the row and column selectors.
- Enter Matrix Values: Fill in the matrix cells with your numerical values. Use the Random or Identity buttons for quick setup.
- Calculate: Click the Calculate button to perform the selected operation.
- Review Results: View the results in the calculation panel. Copy results or save them as a new matrix.
- Try Examples: Use the Quick Examples to learn different matrix operations with pre-loaded examples.
Available Matrix Operations
Basic Operations
- Addition (A + B): Add two matrices of same dimensions
- Subtraction (A - B): Subtract matrix B from A
- Multiplication (A × B): Multiply matrices (cols of A = rows of B)
- Scalar Multiplication (k × A): Multiply matrix by a scalar value
Advanced Operations
- Determinant (det(A)): Calculate determinant of square matrix
- Inverse (A⁻¹): Find inverse of invertible matrix
- Transpose (Aᵀ): Flip matrix over its diagonal
- Rank: Determine matrix rank (linear independence)
Eigen Operations
- Eigenvalues (eig(A)): Find characteristic values
- Trace (tr(A)): Sum of diagonal elements
- Matrix Power (Aⁿ): Raise matrix to integer power
- Linear System (A·X = B): Solve matrix equation
Special Features
- Random Matrix: Generate random matrix values
- Identity Matrix: Create identity matrix quickly
- Precision Control: Adjust decimal precision (2-10 decimals)
- Save Results: Save calculation results as new matrix
Why Use This Matrix Calculator?
Matrix calculations are fundamental to many scientific and engineering disciplines. This calculator offers significant advantages:
Time Saving
Perform complex matrix operations in seconds that would take minutes or hours manually. No more tedious calculations by hand.
Accuracy
Eliminate human calculation errors. The calculator uses precise numerical algorithms with adjustable decimal precision.
Educational
Perfect for students learning linear algebra. Visualize matrix operations and verify homework solutions instantly.
Real-World Applications of Matrix Operations
| Field | Application | Key Operations |
|---|---|---|
| Computer Graphics | 3D transformations, rotations, scaling | Matrix multiplication, inverse, transpose |
| Machine Learning | Neural networks, principal component analysis | Eigenvalues, matrix decomposition |
| Engineering | Structural analysis, circuit analysis | Linear systems, determinants |
| Economics | Input-output models, optimization | Matrix inversion, rank |
| Physics | Quantum mechanics, relativity | Eigenvalues, matrix exponentials |
Matrix Operation Formulas & Rules
Matrix Multiplication Rules
Compatibility: For A × B to be defined:
cols(A) = rows(B)
Result Size: Result matrix C has dimensions:
rows(C) = rows(A), cols(C) = cols(B)
Determinant Formulas
2×2: det = ad - bc
For matrix [[a,b],[c,d]]
3×3: Rule of Sarrus
Sum of diagonal products
Properties: det(AB) = det(A)×det(B)
Multiplicative property
Inverse Matrix Conditions
- Matrix must be square (same rows and columns)
- Determinant must be non-zero: det(A) ≠ 0
- Matrix must be non-singular (full rank)
- Inverse formula for 2×2: A⁻¹ = (1/det(A)) × [[d,-b],[-c,a]]
Frequently Asked Questions (FAQ)
What is the maximum matrix size supported?
This calculator supports matrices up to 5×5 for optimal performance and readability. While larger matrices can be handled theoretically, 5×5 provides a good balance between computational capability and user interface clarity. For operations requiring larger matrices, specialized mathematical software may be more appropriate.
How accurate are the calculations?
The calculator uses JavaScript's double-precision floating-point arithmetic (64-bit), providing approximately 15-17 decimal digits of precision. You can adjust the decimal display precision from 2 to 10 decimals. For operations like matrix inversion and eigenvalue calculation, numerical stability is maintained through robust algorithms, though extremely ill-conditioned matrices may show numerical instability.
Can I solve systems of linear equations?
Yes! Use the "A·X = B" operation to solve linear systems. Enter the coefficient matrix as A and the constant vector as B. The calculator will attempt to solve for X. For square coefficient matrices with non-zero determinant, it uses matrix inversion. For non-square or singular matrices, it provides the best solution using numerical methods.
What are eigenvalues and eigenvectors used for?
Eigenvalues and eigenvectors are fundamental in many applications: principal component analysis (PCA) in statistics, stability analysis in engineering, quantum mechanics in physics, and Google's PageRank algorithm. Eigenvalues represent scaling factors, while eigenvectors represent directions that remain unchanged under the linear transformation represented by the matrix.
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