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 $\mathbf C$:
- $\adj {\mathbf A} = \mathbf C^\intercal$
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:
- $\adj {\mathbf A} = \begin {pmatrix} A_{1 1} & A_{2 1} & A_{3 1} \\ A_{1 2} & A_{2 2} & A_{3 2} \\ A_{1 3} & A_{2 3} & A_{3 3} \end {pmatrix}$
Also see
- Results about adjugate matrices can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): adjoint: 2.
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.7$ Minors and cofactors
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): adjoint (of a matrix)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): adjoint (of a matrix)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): adjoint
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): adjoint (adjugate) (of a matrix)