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 =
  -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]
[ -4, 2*x - z,     2*z]
[ -1,       y, x - 2*y]

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 =
[  d/(a*d - b*c), -b/(a*d - b*c)]
[ -c/(a*d - b*c),  a/(a*d - b*c)]

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 matrix variable

Square matrix, specified as a numeric matrix, matrix of symbolic scalar variables, or symbolic matrix variable (since R2021a).

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.