Matrix Calculator - Linear Algebra Operations Online
Perform matrix operations online with our free matrix calculator. Calculate matrix addition, subtraction, multiplication, determinant, inverse, and transpose instantly. Perfect for students, engineers, and anyone working with linear algebra.
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 applications in physics, engineering, computer graphics, economics, and data science. Each element in a matrix is identified by its row and column position.
Matrix Operations Explained
- Matrix Addition: Add corresponding elements of two matrices with the same dimensions. If A and B are m×n matrices, then C = A + B where c[i][j] = a[i][j] + b[i][j].
- Matrix Subtraction: Subtract corresponding elements of two matrices with the same dimensions. Similar to addition but subtracting elements: c[i][j] = a[i][j] - b[i][j].
- Matrix Multiplication: Multiply two matrices where the number of columns in the first matrix equals the number of rows in the second. The result is a new matrix with dimensions (rows of first) × (columns of second).
- Determinant: A scalar value calculated from a square matrix that provides important properties about the matrix. Non-zero determinant indicates the matrix is invertible.
- Matrix Inverse: For a square matrix A, the inverse A⁻¹ satisfies A × A⁻¹ = I (identity matrix). Only exists when determinant is non-zero.
- Matrix Transpose: Flip a matrix over its diagonal, converting rows to columns and columns to rows. If A is m×n, then Aᵀ is n×m.
Matrix Dimensions and Compatibility
Addition and Subtraction:
Both matrices must have identical dimensions (same number of rows and columns). You cannot add a 2×3 matrix to a 3×2 matrix.
Multiplication:
For A × B to be valid, the number of columns in A must equal the number of rows in B. If A is m×n and B is n×p, the result will be m×p.
Square Matrices:
Determinant and inverse operations require square matrices (same number of rows and columns). Common sizes include 2×2, 3×3, and 4×4 matrices.
Common Matrix Applications
- Computer Graphics: Transformations (rotation, scaling, translation) in 2D and 3D graphics
- Physics: Quantum mechanics, mechanics, and electromagnetic field calculations
- Engineering: Structural analysis, circuit analysis, and control systems
- Economics: Input-output models, optimization problems, and econometrics
- Data Science: Machine learning algorithms, principal component analysis (PCA)
- Statistics: Covariance matrices, regression analysis, and multivariate analysis
- Cryptography: Encryption algorithms and coding theory
Matrix Properties
- Identity Matrix: Square matrix with 1s on the diagonal and 0s elsewhere. Acts as multiplicative identity (A × I = A).
- Zero Matrix: Matrix with all elements equal to zero. Acts as additive identity (A + 0 = A).
- Symmetric Matrix: Square matrix equal to its transpose (A = Aᵀ). Common in physics and statistics.
- Diagonal Matrix: Square matrix with non-zero elements only on the main diagonal.
- Orthogonal Matrix: Square matrix where A × Aᵀ = I. Represents rotations and reflections.
How to Use the Matrix Calculator
- Set the dimensions for Matrix A (rows × columns)
- Enter values for Matrix A in the input grid
- Set dimensions and values for Matrix B if needed
- Use quick-fill buttons (Clear, All 0, All 1, Identity) for convenience
- Select the desired operation (addition, multiplication, etc.)
- View the result matrix instantly
Matrix Multiplication Example
To multiply a 2×3 matrix A by a 3×2 matrix B:
A = [[1, 2, 3], [4, 5, 6]] (2×3)
B = [[7, 8], [9, 10], [11, 12]] (3×2)
Result = [[58, 64], [139, 154]] (2×2)
Each element in the result is calculated by taking the dot product of the corresponding row from A and column from B.
Frequently Asked Questions
What is the maximum matrix size supported?
Our calculator supports matrices up to 10×10. This covers most practical applications while maintaining fast calculations and easy visualization.
Can I multiply matrices of different sizes?
Yes, but only if the number of columns in the first matrix equals the number of rows in the second matrix. For example, you can multiply a 2×3 matrix by a 3×4 matrix, resulting in a 2×4 matrix.
Why can't I calculate the inverse of my matrix?
Matrix inverse only exists for square matrices (same rows and columns) with a non-zero determinant. If your matrix is singular (determinant = 0), it has no inverse.
What is the identity matrix?
The identity matrix is a square matrix with 1s on the main diagonal and 0s everywhere else. It acts as the multiplicative identity, meaning A × I = I × A = A for any compatible matrix A.
How is matrix multiplication different from element-wise multiplication?
Matrix multiplication follows specific rules where each element in the result is the dot product of a row and column. Element-wise multiplication (Hadamard product) simply multiplies corresponding elements, requiring matrices of the same size.
What does the determinant tell us?
The determinant provides information about the matrix's properties. A non-zero determinant means the matrix is invertible and represents a transformation that doesn't collapse space. The absolute value represents the scaling factor of the transformation.
Can I use decimal numbers in the matrix?
Yes! Our calculator supports decimal numbers (floating-point values). You can enter values like 1.5, -3.14, or 0.001 in any matrix cell.