# Inverse Matrix Calculator

URL: https://speedycalculator.com/calculators/math/matrix-inverse/
Last updated: 2026-10-30

> The inverse matrix calculator finds A⁻¹ for a 2 × 2, 3 × 3 or 4 × 4 matrix with exact fractions when the entries are integers, and reports when the matrix is singular.

## Inputs

- Matrix size
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄

## Results

- Inverse A⁻¹
- Determinant |A|
- Check
- A⁻¹ as decimals

## Formula

```
A⁻¹ = adj(A) / det(A)
2 × 2: [[a, b], [c, d]]⁻¹ = 1/(ad − bc) · [[d, −b], [−c, a]]
```

## Explanation

The inverse undoes a matrix: A⁻¹A = I. It solves systems of equations (x = A⁻¹b), reverses transformations in graphics and appears throughout statistics and engineering.

### How to use the Inverse Matrix Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The calculator computes the determinant and the adjugate; dividing gives the inverse. A matrix with determinant 0 is singular and has no inverse.

## Worked example

Inputs — Matrix size: 2 × 2; a₁: 4; b₁: 7; a₂: 2; b₂: 6.

Results — **Inverse A⁻¹ = ⎡ 3/5 −7/10 ⎤
⎣ −1/5 2/5 ⎦**; Determinant |A| = 10; Check = A × A⁻¹ = I ✓; A⁻¹ as decimals = ⎡ 0.6 −0.7 ⎤
⎣ −0.2 0.4 ⎦.

## Key terms

- **Singular matrix**: A square matrix with determinant 0.

## FAQ

**When does a matrix have an inverse?**
When its determinant is non-zero.

**What is the inverse of [[4, 7], [2, 6]]?**
1/10 · [[6, −7], [−2, 4]].

**Is (AB)⁻¹ = A⁻¹B⁻¹?**
No – (AB)⁻¹ = B⁻¹A⁻¹.

**Should I invert a matrix to solve Ax = b?**
For hand work yes; numerically, elimination is faster and more accurate.


## Sources

- [Wolfram MathWorld – Matrix inverse](https://mathworld.wolfram.com/MatrixInverse.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)