Math 2.0

Linear and multilinear algebra; matrix theory Definition

Adjugate matrix

Builds on CofactorTranspose

Definition

Let KK be a commutative ring, let n≥2n \ge 2, and let AA be an n×nn \times n matrix over KK with cofactors CijC_{ij}. The adjugate of AA is the n×nn \times n matrix adj⁡(A)\operatorname{adj}(A) with entries adj⁡(A)ij=Cji.\operatorname{adj}(A)_{ij} = C_{ji}. So adj⁡(A)\operatorname{adj}(A) is the transpose of the matrix of cofactors (Cij)(C_{ij}).

Remarks

For n=2n = 2, adj⁡(abcd)=(d−b−ca)\operatorname{adj} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. The adjugate satisfies Aadj⁡(A)=det⁡(A)InA \operatorname{adj}(A) = \det(A) I_n. Older texts call it the classical adjoint; it is not the adjoint of an inner product space.

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  1. Web https://en.wikipedia.org/wiki/Adjugate_matrix, Definition
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Tag0GMPYM0NSR
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WritingW1 stated
MSC15A09, 15A15
DefinesAdjugate matrix
UsesCofactor, Transpose

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