adjoint

Classical adjoint (adjugate) of square matrix

Syntax

X = adjoint(A)

Description

X = adjoint(A) returns the Classical Adjoint (Adjugate) Matrix X of A, such that A*X = det(A)*eye(n) = X*A, where n is the number of rows in A.

Examples

Classical Adjoint (Adjugate) of Matrix

Find the classical adjoint of a numeric matrix.

A = magic(3);
X = adjoint(A)

X = 3×3

  -53.0000   52.0000  -23.0000
   22.0000   -8.0000  -38.0000
    7.0000  -68.0000   37.0000

Find the classical adjoint of a symbolic matrix.

syms x y z
A = sym([x y z; 2 1 0; 1 0 2]);
X = adjoint(A)

X = 
$$\left(\begin{array}{ccc}2& -2\hspace{0.17em}y& -z\\ -4& 2\hspace{0.17em}x-z& 2\hspace{0.17em}z\\ -1& y& x-2\hspace{0.17em}y\end{array}\right)$$

Verify that det(A)*eye(3) = X*A by using isAlways.

cond = det(A)*eye(3) == X*A;
isAlways(cond)

ans = 3×3 logical array

   1   1   1
   1   1   1
   1   1   1

Compute Inverse Using Classical Adjoint and Determinant

Compute the inverse of this matrix by computing its classical adjoint and determinant.

syms a b c d
A = [a b; c d];
invA = adjoint(A)/det(A)

invA = 
$$\left(\begin{array}{cc}\frac{d}{a\hspace{0.17em}d-b\hspace{0.17em}c}& -\frac{b}{a\hspace{0.17em}d-b\hspace{0.17em}c}\\ -\frac{c}{a\hspace{0.17em}d-b\hspace{0.17em}c}& \frac{a}{a\hspace{0.17em}d-b\hspace{0.17em}c}\end{array}\right)$$

Verify that invA is the inverse of A.

isAlways(invA == inv(A))

ans = 2×2 logical array

   1   1
   1   1

Input Arguments

A — Square matrix

numeric matrix | matrix of symbolic scalar variables | symbolic function | symbolic matrix variable | symbolic matrix function | symbolic expression

Square matrix, specified as a numeric matrix, matrix of symbolic scalar variables, symbolic matrix variable, symbolic function, symbolic matrix function, or symbolic expression.

Data Types: single | double | sym | symfun | symmatrix | symfunmatrix

More About

Classical Adjoint (Adjugate) Matrix

The classical adjoint, or adjugate, of a square matrix A is the square matrix X, such that the (i,j)-th entry of X is the (j,i)-th cofactor of A.

The (j,i)-th cofactor of A is defined as follows.

$$a_{ji}{}^{\prime }={\left(-1\right)}^{i+j}\det \left(A_{ij}\right)$$

A_ij is the submatrix of A obtained from A by removing the i-th row and j-th column.

The classical adjoint matrix should not be confused with the adjoint matrix. The adjoint is the conjugate transpose of a matrix while the classical adjoint is another name for the adjugate matrix or cofactor transpose of a matrix.

Version History

Introduced in R2013a

R2022a: Compute classical adjoint of symbolic matrix functions

The adjoint function accepts an input argument of type symfunmatrix.

R2021a: Compute classical adjoint of symbolic matrix variables

The adjoint function accepts an input argument of type symmatrix.

R2018b: Compute classical adjoint of numeric matrices

The adjoint function accepts a numeric matrix as an input argument.

The adjoint function supports numeric matrices of type double and single, as well as symbolic matrices of type sym and symfun.