adjoint

Adjoint of symbolic square matrix

Syntax

X = adjoint(A)

Description

X = adjoint(A) returns the adjoint matrix X of A. The adjoint of a matrix A is the matrix X, such that A*X = det(A)*eye(n) = X*A, where n is the number of rows in A and eye(n) is the n-by-n identity matrix.

Input Arguments

A

Symbolic square matrix.

Output Arguments

X

Symbolic square matrix of the same size as A.

Examples

Compute the adjoint of this 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 A*X = det(A)*eye(3), where eye(3) is the 3-by-3 identity matrix:

isAlways(A*X == det(A)*eye(3))
ans =
     1     1     1
     1     1     1
     1     1     1

Also verify that det(A)*eye(3) = X*A:

isAlways(det(A)*eye(3) == X*A)
ans =
     1     1     1
     1     1     1
     1     1     1

Compute the inverse of this matrix by computing its 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 =
     1     1
     1     1

More About

expand all

Adjoint of a Square Matrix

The adjoint 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.

Cofactor of a Matrix

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

aji=(1)i+jdet(Aij)

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

See Also

| | |

Was this topic helpful?