Inverse Matrix Calculator
Invert square matrices with the 2×2 formula, Gauss-Jordan, or the adjugate method.
Inverse Matrix Calculator
Calculate matrix inverses with step-by-step solutions using multiple methods
Matrix Input
Matrix Properties
About Matrix Inverses
What an inverse is
For a square matrix A, the inverse A⁻¹ satisfies A A⁻¹ = A⁻¹ A = I (identity). Only matrices with det(A) ≠ 0 have one. Those are non-singular; det = 0 means singular—no inverse.
Main use: Ax = b becomes x = A⁻¹b when A is invertible. In numerical work you often solve via LU/QR instead of forming A⁻¹ explicitly, but the concept still matters.
Methods
2×2 direct formula
A = [[a,b],[c,d]] → A⁻¹ = (1/(ad−bc)) [[d,−b],[−c,a]]
Fastest for 2×2 when the determinant is nonzero.
Gauss-Jordan
Augment [A | I], row-reduce to [I | A⁻¹]. Works for any size; same idea as solving systems by elimination.
Adjugate
A⁻¹ = (1/det(A)) adj(A)
Cofactors → transpose → scale. Useful for 3×3 by hand; expensive for larger n.
Useful identities
- (AB)⁻¹ = B⁻¹ A⁻¹
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ
- (A⁻¹)⁻¹ = A
- det(A⁻¹) = 1/det(A)
- Orthogonal: A⁻¹ = Aᵀ
Frequently Asked Questions
When is there no inverse?
When det(A) = 0—rows or columns linearly dependent. The map collapses dimension and cannot be undone uniquely.
Which method should I use?
2×2: direct formula. 3×3 by hand: adjugate or Gauss-Jordan. Larger or production code: factorization (LU with pivoting), not a hand adjugate.
How do I check the answer?
Multiply A × A⁻¹. You want I. Tiny off-diagonal junk (~1e−15) is floating-point noise.
Singular vs non-singular?
Non-singular: invertible, det ≠ 0. Singular: det = 0, no inverse, no unique solution for Ax = b in general.
Why avoid forming inverses in software?
Cost and stability. Solving Ax = b via decomposition is usually cheaper and better conditioned than computing A⁻¹ then multiplying.
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