Perform matrix operations including addition, subtraction, multiplication, determinant, inverse, and transpose calculations.
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are fundamental in linear algebra and have numerous applications in mathematics, physics, engineering, computer science, and data analysis.
Matrix addition and subtraction can only be performed on matrices of the same dimensions. Each corresponding element is added or subtracted:
Example: If A = [1, 2; 3, 4] and B = [5, 6; 7, 8], then A + B = [6, 8; 10, 12]
Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix. The result is a matrix where each element is the dot product of the corresponding row and column.
Formula: (AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ
The determinant is a scalar value that can be computed from a square matrix. It provides important information about the matrix, such as whether it's invertible.
For 2×2 matrix: det([a, b; c, d]) = ad - bc
The inverse of a matrix A is denoted A⁻¹ and satisfies AA⁻¹ = A⁻¹A = I (identity matrix). Only square matrices with non-zero determinants have inverses.
The transpose of a matrix is formed by swapping rows and columns. If A is an m×n matrix, then Aᵀ is an n×m matrix where (Aᵀ)ᵢⱼ = Aⱼᵢ.
Scalar multiplication involves multiplying every element of a matrix by a scalar (constant) value.
Matrices are used for 2D and 3D transformations including rotation, scaling, translation, and projection in computer graphics and game development.
Matrices provide an efficient way to solve systems of linear equations using methods like Gaussian elimination or matrix inversion.
In statistics and machine learning, matrices represent datasets, covariance matrices, and transformation matrices for principal component analysis.
Adjacency matrices represent relationships in graphs and networks, useful in social network analysis and routing algorithms.
| Matrix Type | Description | Properties |
|---|---|---|
| Identity Matrix | Square matrix with 1s on diagonal, 0s elsewhere | AI = IA = A |
| Zero Matrix | All elements are zero | A + 0 = A |
| Diagonal Matrix | Non-zero elements only on main diagonal | Easy to compute powers |
| Symmetric Matrix | A = Aᵀ | Real eigenvalues |
Our matrix calculator supports all fundamental matrix operations with an intuitive interface. Enter your matrices and select the desired operation to get instant results. The calculator handles matrices up to 5×5 and provides step-by-step solutions for better understanding.