Definition:Adjugate Matrix

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Not to be confused with Definition:Adjoint Matrix.

Definition

Let $R$ be a commutative ring with unity.

Let $\mathbf A \in R^{n \times n}$ 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 this as the adjoint matrix of $\mathbf A$.

However, as this term 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 $3$:

$\mathbf A = \begin {pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end {pmatrix}$

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


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

$\adj {\mathbf A} = \begin {pmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{23} \\ A_{13} & A_{23} & A_{33} \end {pmatrix}$


Also see

  • Results about adjugate matrices can be found here.


Sources