Not to be confused with Definition:Adjoint Matrix.

## Definition

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $\mathbf C$ be its cofactor matrix.

The adjugate matrix of $\mathbf A$ is the transpose of $\mathbf C$:

$\adj {\mathbf A} = \mathbf C^\intercal$

## Also known as

Some sources refer to the adjugate matrix of $\mathbf A$ as the adjoint matrix of $\mathbf A$.

The use of adjugate may be less common than that of adjoint.

However, as adjoint matrix is also used for the Hermitian conjugate, to avoid ambiguity it is recommended that it not be used.

## Examples

### $2 \times 2$ Square Matrix

Let $\mathbf A$ be the square matrix of order $2$:

$\mathbf A = \begin {pmatrix} a & b \\ c & d \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} d & -b \\ -c & a \end {pmatrix}$

### $3 \times 3$ Square Matrix

Let $\mathbf A$ be the square matrix of order $3$:

$\mathbf A = \begin {pmatrix} a_{1 1} & a_{1 2} & a_{1 3} \\ a_{2 1} & a_{2 2} & a_{2 3} \\ a_{3 1} & a_{3 2} & a_{3 3} \end {pmatrix}$

Let $A_{i j}$ denote the cofactor of element $a_{ij}$.

Then the adjugate matrix of $\mathbf A$ is:

$\adj {\mathbf A} = \begin {pmatrix} A_{1 1} & A_{2 1} & A_{3 1} \\ A_{1 2} & A_{2 2} & A_{3 2} \\ A_{1 3} & A_{2 3} & A_{3 3} \end {pmatrix}$