Adjoint Matrix

 

https://en.wikipedia.org/wiki/Adjugate_matrix

 

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.^[1]

The adjugate^[2] has sometimes been called the "adjoint",^[3] but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

 

Contents

  [hide] 

• 1Definition
• 2Examples
  • 2.11 × 1 generic matrix
  • 2.22 × 2 generic matrix
  • 2.33 × 3 generic matrix
  • 2.43 × 3 numeric matrix
• 3Properties
  • 3.1Inverses
  • 3.2Characteristic polynomial
  • 3.3Jacobi's formula
  • 3.4Cayley–Hamilton formula
• 4See also
• 5References
• 6External links

 

Definition[edit]

The adjugate of A is the transpose of the cofactor matrix C of A,

{\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}~.}

In more detail, suppose R is a commutative ring and A is an n×n matrix with entries from R.

• The (i,j) minor of A, denoted M_ij, is the determinant of the (n − 1)×(n − 1) matrix that results from deleting row i and column j of A.
• The cofactor matrix of A is the n×n matrix C whose (i, j) entry is the (i, j) cofactor of A,

{\displaystyle \mathbf {C} _{ij}=(-1)^{i+j}\mathbf {M} _{ij}~.}

• The adjugate of A is the transpose of C, that is, the n×n matrix whose (i,j) entry is the (j,i) cofactor of A,

{\displaystyle \operatorname {adj} (\mathbf {A} )_{ij}=\mathbf {C} _{ji}=(-1)^{i+j}\mathbf {M} _{ji}\,}.

The adjugate is defined as it is so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are det(A),

{\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\det(\mathbf {A} )\,\mathbf {I} ~.}

A is invertible if and only if det(A) is an invertible element of R, and in that case the equation above yields

{\displaystyle \operatorname {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{-1}~,}

{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\operatorname {adj} (\mathbf {A} )~.}