How to Use the Matrix Calculator

Select a matrix size (2×2 or 3×3) and choose an operation. Enter the values for each matrix element and click Calculate to see the result.

Matrix Operations

Addition and Subtraction

Matrix addition and subtraction are performed element by element. Both matrices must have the same dimensions. The result is a matrix of the same size.

Multiplication

Matrix multiplication combines rows of the first matrix with columns of the second. Each element in the result is the dot product of a row from A and a column from B. Note that matrix multiplication is not commutative: A×B ≠ B×A in general.

Determinant

The determinant is a scalar value that provides important information about the matrix:

If det(A) ≠ 0, the matrix is invertible
If det(A) = 0, the matrix is singular (no inverse exists)
The absolute value relates to the scaling factor of the linear transformation

Transpose

The transpose of a matrix is obtained by flipping it over its main diagonal, effectively converting rows to columns and vice versa. If A is an m×n matrix, its transpose Aᵀ is an n×m matrix.

Inverse

The inverse of a matrix A (denoted A⁻¹) is a matrix such that A × A⁻¹ = I, where I is the identity matrix. A matrix has an inverse only if its determinant is non-zero.

Applications of Matrices

Computer graphics and 3D transformations
Solving systems of linear equations
Data analysis and machine learning
Physics simulations and engineering
Economics and financial modeling
Network analysis and graph theory