# Adjoint Matrix Calculator

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

> The adjoint matrix calculator finds the adjugate (classical adjoint) of a 2 × 2, 3 × 3 or 4 × 4 matrix – the transpose of its cofactor matrix – and uses it to give the inverse.

## Inputs

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

## Results

- Adjoint (adjugate) matrix
- Determinant
- Inverse A⁻¹ = adj(A)/det(A)

## Formula

```
adj(A) = Cᵀ, where Cᵢⱼ = (−1)^(i+j) · Mᵢⱼ
A⁻¹ = adj(A) / det(A)
```

## Explanation

The adjugate packages all the cofactors of a matrix. Its key property, A · adj(A) = det(A) · I, gives a formula for the inverse and for Cramer's rule. For a 2 × 2 matrix [[a, b], [c, d]], adj(A) = [[d, −b], [−c, a]].

### How to use the Adjoint 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

Each cofactor is a signed minor – the determinant of the matrix with one row and one column removed. The cofactor matrix is transposed to give the adjugate; dividing by the determinant gives the inverse.

## Worked example

Inputs — Matrix size: 3 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; a₃: 2; b₃: 0; c₃: 5.

Results — **Adjoint (adjugate) matrix = ⎡ 15 −5 −6 ⎤
⎢ −5 16 2 ⎥
⎣ −6 2 11 ⎦**; Determinant = 43; Inverse A⁻¹ = adj(A)/det(A) = ⎡ 15/43 −5/43 −6/43 ⎤
⎢ −5/43 16/43 2/43 ⎥
⎣ −6/43 2/43 11/43 ⎦.

## Key terms

- **Cofactor**: A signed minor (−1)^(i+j) Mᵢⱼ.
- **Adjugate**: The transpose of the cofactor matrix.

## FAQ

**What is the adjoint of [[a, b], [c, d]]?**
[[d, −b], [−c, a]].

**Is the adjoint the same as the adjugate?**
In this context yes; "adjoint" also means the conjugate transpose in other contexts.

**What if det(A) = 0?**
The adjugate still exists, but the matrix has no inverse.

**Why transpose the cofactor matrix?**
So that A · adj(A) = det(A) · I.


## Sources

- [Wolfram MathWorld – Adjugate](https://mathworld.wolfram.com/Adjugate.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/)