Definition:Adjugate Matrix

This page is about Adjugate Matrix. For other uses, see Adjugate.

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 $\mathbf C$:

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

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.

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:

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

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:

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