Classical adjoint (adjugate) of square matrix
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]
det(A)*eye(3) = X*A by using
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)]
invA is the inverse of
isAlways(invA == inv(A))
ans = 2×2 logical array 1 1 1 1
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).
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.
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.