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Description

example

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

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Find the classical adjoint of a numeric matrix.

A = magic(3);
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 =
[  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 the inverse of this matrix by computing its classical adjoint and determinant.

syms a b c d
A = [a b; c d];
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

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Square matrix, specified as a numeric or symbolic matrix.

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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.

${a}_{ji}{}^{\prime }={\left(-1\right)}^{i+j}\mathrm{det}\left({A}_{ij}\right)$

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.