Perform matrix operations: add, subtract, multiply, find transpose, determinant, and inverse. Supports 2x2, 3x3, and 4x4 matrices. Results shown with full working.
Matrix multiplication: the element at row i, column j of the result equals the dot product of row i from Matrix A and column j from Matrix B. This only works when A columns equal B rows. For square matrices, this always applies.
The determinant is a scalar value calculated from a square matrix. For 2x2: det = ad - bc. It indicates whether the matrix is invertible (non-zero determinant) and appears in many physics and engineering formulas.
The inverse A⁻¹ satisfies A × A⁻¹ = I (the identity matrix). It only exists when the determinant is non-zero. Used to solve systems of linear equations: Ax = b → x = A⁻¹b.
A matrix is singular when its determinant equals zero — it has no inverse. Geometrically, it means the transformation collapses space into a lower dimension (rows or columns are linearly dependent).
The transpose flips a matrix over its main diagonal — rows become columns and columns become rows. Element A[i][j] becomes A[j][i]. Transpose is used in linear algebra, statistics, and machine learning.
Matrices represent linear transformations, solve systems of equations, describe 3D rotations in computer graphics, model neural network weights, handle data in machine learning, and underpin physics equations.
Only if the number of columns in the first matrix equals the number of rows in the second. For m×n × n×p → m×p result. This calculator handles square matrices up to 4×4.