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.

example

Examples

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Classical Adjoint (Adjugate) of Matrix

Open Live Script

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 = 
 $(\begin{array}{c} 2 & - 2 y & - z \\ - 4 & 2 x - z & 2 z \\ - 1 & y & x - 2 y \end{array})$

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

Open Live Script

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 = 
 $(\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{a d - b c} \end{array})$

Verify that invA is the inverse of A.

isAlways(invA == inv(A))

ans = 2×2 logical array

   1   1
   1   1

Input Arguments

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`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.

More About

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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.

$a_{j i}^{'} = {(- 1)}^{i + j} \det (A_{i j})$

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

expand all

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.

adjoint

Syntax

Description

Examples

Classical Adjoint (Adjugate) of Matrix

Compute Inverse Using Classical Adjoint and Determinant

Input Arguments

A — Square matrix numeric matrix | matrix of symbolic scalar variables | symbolic function | symbolic matrix variable | symbolic matrix function | symbolic expression

More About

Classical Adjoint (Adjugate) Matrix

Version History

R2022a: Compute classical adjoint of symbolic matrix functions

R2021a: Compute classical adjoint of symbolic matrix variables

R2018b: Compute classical adjoint of numeric matrices

See Also

`A` — Square matrix
numeric matrix | matrix of symbolic scalar variables | symbolic function | symbolic matrix variable | symbolic matrix function | symbolic expression