Matrix Inverse Calculator
The inverse of a square matrix A is another matrix A^(-1) such that A * A^(-1) = I (the identity matrix). Only matrices with non-zero determinant (non-singular matrices) have inverses. This calculator computes the inverse for 2x2 and 3x3 matrices and checks whether your matrix is singular (non-invertible) or regular (invertible). The inverse is essential for solving systems of linear equations and reversing linear transformations.
Matrix A
A^(-1)
Matrix inverse formulas
A * A^(-1) = I
2x2: A^(-1) = (1/det) * [[d, -b], [-c, a]]
3x3: Uses adjugate matrix divided by determinant
Matrix inverse calculator: frequently asked questions
What is the inverse of a matrix?
The inverse of a square matrix A, denoted A^(-1), is the matrix that, when multiplied by A, gives the identity matrix. So A * A^(-1) = I. Only square matrices with non-zero determinant have inverses. Singular matrices (det = 0) do not have inverses.
How do you find the inverse of a 2x2 matrix?
For [[a, b], [c, d]], first calculate det = ad - bc. If det ≠ 0, then A^(-1) = (1/det) * [[d, -b], [-c, a]]. Swap the diagonal elements, negate the off-diagonal elements, and divide by the determinant.
How do you find the inverse of a 3x3 matrix?
For a 3x3 matrix, compute the matrix of cofactors, transpose it to get the adjugate matrix, then divide by the determinant. This is more complex than the 2x2 case but follows the same principle.
When does a matrix not have an inverse?
A matrix does not have an inverse if it is singular, which means its determinant is zero. This occurs when the rows or columns are linearly dependent, or when the matrix represents a transformation that collapses space into a lower dimension.
What are applications of matrix inverses?
Matrix inverses are used to solve systems of linear equations, in computer graphics for reversing transformations, in cryptography, and in engineering simulations. Whenever you need to undo a linear transformation, you use the inverse matrix.
Official sources
- Khan Academy: Matrix inverse.
- Wolfram MathWorld: Matrix inverse.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.