Adjugate matrix

In linear algebra, the adjugate or classical adjoint of a square matrix is a matrix which plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions.

The adjugate has sometimes been called the "adjoint", but that terminology is ambiguous. Today, "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

Definition

Suppose R is a commutative ring and A is an n×n matrix with entries from R. Define the (i,j)-minor M_ij of A as the determinant of the (n − 1)×(n − 1) matrix that results from deleting row i and column j of A, and the i,j cofactor of A as

$C_{ij} = (-1)^{i+j} M_{ij} \,$ .

Then the adjugate of A is the n×n matrix whose i,j entry is

$\mathrm{adj}(A)_{ij} = C_{ji} \,$ .

That is, the adjugate of A is the transpose of the "cofactor matrix" (C_ij) of A.

It may (or may not) be helpful to attach names to the steps in the process. You can let M^~_ij be the (n-1) x (n-1) matrix minor, that is, the matrix that results from deleting row i and column j of A. Then M_ij = det( M^~_ij). Let cof(A) be the cofactor matrix mentioned above. Then adj(A) = transpose of cof(A).

Examples

As an example, we have

$\operatorname{adj}\begin{pmatrix} 2& 1&1\\ 0&-1&2\\ 0&2&-1 \end{pmatrix}= \begin{pmatrix} -3&3&3\\ 0&-2&-4\\ 0&-4&-2 \end{pmatrix}$ .

Here, the −4 in the last row, second column was computed by deleting the second row and the last column of the original matrix and computing

$(-1)^{3+2}\;\operatorname{det}\begin{pmatrix}2&1\\ 0&2 \end{pmatrix}=(-1)(4)=-4$ .

A more generic example is this: given matrix

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

its adjugate is

$\mbox{adj}(A) = \begin{pmatrix} +\left| \begin{matrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{matrix} \right| & -\left| \begin{matrix} A_{12} & A_{13} \\ A_{32} & A_{33} \end{matrix} \right| & +\left| \begin{matrix} A_{12} & A_{13} \\ A_{22} & A_{23} \end{matrix} \right| \\ & & \\ -\left| \begin{matrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{matrix} \right| & +\left| \begin{matrix} A_{11} & A_{13} \\ A_{31} & A_{33} \end{matrix} \right| & -\left| \begin{matrix} A_{11} & A_{13} \\ A_{21} & A_{23} \end{matrix} \right| \\ & & \\ +\left| \begin{matrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{matrix} \right| & -\left| \begin{matrix} A_{11} & A_{12} \\ A_{31} & A_{32} \end{matrix} \right| & +\left| \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} \right| \end{pmatrix}$

where

$\left| \begin{matrix} A_{ij} & A_{kl} \\ A_{mn} & A_{op} \end{matrix} \right|=\det\left( \begin{matrix} A_{ij} & A_{kl} \\ A_{mn} & A_{op} \end{matrix} \right);$

while:

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

Applications

As a consequence of Laplace's formula for the determinant of an n×n matrix A, we have

$(*) \qquad A\, \mathrm{adj}(A) = \mathrm{adj}(A)\, A = \det(A)\, I$

where I is the n×n identity matrix. Indeed, the i,i entry of the product A adj(A) is the scalar product of row i of A with row i of (C_ij), which is simply the Laplace formula for det(A) expanded by row i, and for i ≠ j the i,j entry of the product is the scalar product of row i of A with row j of (C_ij), which is the Laplace formula for the determinant of a matrix whose i and j rows are equal and is therefore zero.

From this formula follows one of the most important results in matrix algebra: A matrix A over a commutative ring R is invertible if and only if det(A) is invertible in R.

For if A is an invertible matrix then 1 = det(I) = det(A A⁻¹) = det(A) det(A⁻¹), and if det(A) is a unit then (*) above shows that

$A^{-1} = \det(A)^{-1}\, \mathrm{adj}(A)$ .

Properties

The adjugate has the properties

$\mathrm{adj}(I) = I\,$

$\mathrm{adj}(AB) = \mathrm{adj}(B)\,\mathrm{adj}(A)\,$

for all n×n matrices A and B.

The adjugate preserves transposition:

$\mathrm{adj}(A^T) = \mathrm{adj}(A)^T\,$ .

Furthermore,

$\det\big(\mathrm{adj}(A)\big) = \det(A)^{n-1}\,$ .

If p(t) = det(A − tI) is the characteristic polynomial of A and we define the polynomial q(t) = (p(0) − p(t))/t, then

$\mathrm{adj}(A) = q(A) = -(p_1 I + p_2 A + p_3 A^2 + \cdots + p_{n} A^{n-1})$ ,

where $p j$ are the coefficients of p(t),

$p(t) = p_0 + p_1 t + p_2 t^2 + \cdots p_{n} t^{n}.$

The adjugate also appears in the formula of the derivative of the determinant.

External links

Matrix Reference Manual

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Adjugate matrix

Definition

Examples

Applications

Properties

External links

Adjugate matrix

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