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adjoint

Classical adjoint (adjugate) of square matrix

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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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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 = 

(2-2 y-z-42 x-z2 z-1yx-2 y)

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

cond = det(A)*eye(3) == X*A;
isAlways(cond)
ans = 3x3 logical array

   1   1   1
   1   1   1
   1   1   1
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 = 

(da d-b c-ba d-b c-ca d-b caa d-b c)

Verify that invA is the inverse of A.

isAlways(invA == inv(A))
ans = 2x2 logical array

   1   1
   1   1

Input Arguments

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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

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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.

aji′=(−1)i+jdet(Aij)

Aij 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

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See Also

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