A⁻¹ Inverse Matrix Calculator

Inverse Matrix Calculator

The inverse of a square matrix A is the matrix A⁻¹ such that A·A⁻¹ = I, and this calculator finds it using Gauss-Jordan elimination with every row operation shown. Verifies A × A⁻¹ = I automatically.

A⁻¹

Inverse Matrix Calculator

Gauss-Jordan method with complete steps

✓ Gauss-Jordan✓ Verification✓ Free⚡ Loading precision 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

  1. Form the augmented matrix [A | I] where I is the identity matrix
  2. Apply row operations to reduce the left side to the identity: [I | A⁻¹]
  3. 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)

Worked Example

For A = [[2,1],[1,1]]: the determinant is (2×1) - (1×1) = 1. Since the determinant isn't zero, the inverse exists: A⁻¹ = [[1,-1],[-1,2]]. You can check this is correct because A × A⁻¹ gives the identity matrix [[1,0],[0,1]].

When Does a Matrix Have No Inverse?

A matrix has no inverse when its determinant is exactly zero — this is called a "singular" matrix. This happens when one row (or column) can be created by combining the others, meaning the matrix doesn't hold enough independent information to be reversed.

Applications

  • Solving systems of linear equations directly: x = A⁻¹b
  • Cryptography (encoding and decoding messages using matrix transformations)
  • Computer graphics (reversing a transformation, like undoing a rotation or scale)

Frequently Asked Questions

This happens when the determinant equals zero. Check whether one row or column can be made by adding or scaling the others — if so, the matrix is singular and has no inverse.
No — only square matrices (same number of rows and columns) can have an inverse. Non-square matrices use a related concept called a pseudoinverse instead, which this calculator doesn't currently cover.
Multiply your original matrix by the calculated inverse. If the result is the identity matrix (1s on the diagonal, 0s everywhere else), the inverse is correct.

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