Free matrix calculator: determinant, inverse, transpose, addition, multiplication and rank with steps.
A matrix calculator performs the standard matrix operations: addition, subtraction, multiplication, transpose, inverse, determinant, and eigenvalue computation. Matrices are the fundamental data structure of linear algebra and underpin computer graphics, machine learning, quantum mechanics, and structural engineering.
Hand-computing a 4×4 inverse or determinant is a ~30-minute exercise prone to arithmetic errors. A symbolic calculator gives exact answers instantly for matrices up to size 6×6 — the practical limit for human verification.
A^{-1} = \frac{1}{\det(A)} \text{adj}(A)
Inputs: 2×2 matrix A = [[2, 3], [1, 4]]
A square matrix is non-invertible (singular) when its determinant is zero — meaning its columns are linearly dependent. Geometrically, the matrix "squashes" its input vectors onto a lower-dimensional space, losing information. Singular matrices arise in over-determined or under-determined systems of equations.
In general AB ≠ BA. Multiplication encodes composition of linear transformations, and the order of transformations matters — rotating then scaling is not the same as scaling then rotating. Two specific cases where AB = BA: when A or B is the identity, and when A and B share eigenvectors.
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