Matrix Inverse Calculator
Find the inverse of a 2×2, 3×3, or 4×4 matrix — returns A⁻¹ where A·A⁻¹ = I.
How it works
- 1Enter all entries of your n×n matrix A using the grid, then choose the size (2, 3, or 4).
- 2The calculator applies Gauss-Jordan elimination to the augmented matrix [A | I], reducing the left half to the identity.
- 3If the pivot is effectively zero, the matrix is singular and no inverse exists — the tool reports this clearly.
Use cases
- Solving systems of linear equations by computing x = A⁻¹b instead of row-reducing each time.
- Computer graphics: inverting transformation matrices to reverse rotations, scaling, and translations.
- Statistics and machine learning: computing (XᵀX)⁻¹ in ordinary least-squares regression.
Frequently asked questions
What makes a matrix invertible?
A square matrix is invertible — also called non-singular or full-rank — if and only if its determinant is non-zero. Equivalently, its rows (and columns) must be linearly independent.
What does A · A⁻¹ = I mean?
Multiplying any square matrix by its inverse yields the identity matrix I (ones on the diagonal, zeros elsewhere). This is the matrix equivalent of a number times its reciprocal equalling 1.
Can a non-square matrix have an inverse?
Not in the classical sense. Only square matrices can have a two-sided inverse. Non-square matrices can have a pseudoinverse (Moore-Penrose), but that is a different concept and is not computed here.
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