Matrix Calculator
Perform matrix operations such as addition, subtraction, multiplication, determinant, inverse, transpose, eigenvalues, and more. Enter your matrices and select an operation to see the results.
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About Matrix Operations
A matrix is a rectangular array of numbers arranged in rows and columns. Matrix operations are fundamental in linear algebra and have applications in various fields including computer graphics, engineering, physics, and statistics.
Basic Operations
Matrix Addition: Two matrices can be added if they have the same dimensions. The sum is obtained by adding corresponding elements.
Matrix Subtraction: Similar to addition, two matrices can be subtracted if they have the same dimensions. The difference is obtained by subtracting corresponding elements.
Matrix Multiplication: The product of two matrices A (m×n) and B (n×p) is a new matrix C (m×p) where each element is the dot product of the corresponding row from A and column from B.
Scalar Multiplication: A matrix can be multiplied by a scalar (a single number) by multiplying every element of the matrix by the scalar.
Transpose: The transpose of a matrix is obtained by flipping the matrix over its main diagonal, switching the row and column indices of the matrix.
Advanced Operations
Determinant: The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix.
Inverse: The inverse of a square matrix A is a matrix A⁻¹ such that AA⁻¹ = I, where I is the identity matrix. A matrix must be square and have a non-zero determinant to have an inverse.
Rank: The rank of a matrix is the dimension of the vector space generated by its columns, which equals the maximum number of linearly independent columns of the matrix.
Eigenvalues: An eigenvalue of a square matrix is a scalar λ such that there exists a non-zero vector v (called an eigenvector) where Av = λv.
Trace: The trace of a square matrix is the sum of the elements on its main diagonal.
Applications
Matrix operations are used in solving systems of linear equations, computer graphics transformations, quantum mechanics, statistics (covariance matrices), machine learning (principal component analysis), and many other areas of mathematics and science.