A⁻¹ Inverse Matrix Calculator
Inverse Matrix Calculator
Compute the inverse of any invertible n×n matrix with full Gauss-Jordan elimination. Verifies A × A⁻¹ = I automatically.
Inverse Matrix Calculator
Gauss-Jordan method with complete steps
✓ Gauss-Jordan✓ Verification✓ Free⚡ Loading SymPy Engine…
Matrix A
Quick Examples
How to Find the Inverse Matrix
A square matrix A has an inverse A⁻¹ if and only if det(A) ≠ 0. The inverse satisfies: A × A⁻¹ = A⁻¹ × A = I
Gauss-Jordan Method
- Form the augmented matrix [A | I] where I is the identity matrix
- Apply row operations to reduce the left side to the identity: [I | A⁻¹]
- The right side is the inverse A⁻¹
2×2 Inverse Formula
$$\begin{pmatrix}a & b \\ c & d\end{pmatrix}^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$$
Properties of the Inverse
- (A⁻¹)⁻¹ = A
- (AB)⁻¹ = B⁻¹A⁻¹
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ
- det(A⁻¹) = 1/det(A)