adjoint
Classical adjoint (adjugate) of square matrix
Syntax
Description
returns
the Classical Adjoint (Adjugate) Matrix
X
= adjoint(A
)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 = 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 =
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
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 =
Verify that invA
is the inverse of A
.
isAlways(invA == inv(A))
ans = 2x2 logical array
1 1
1 1
Input Arguments
A
— Square matrix
numeric matrix | matrix of symbolic scalar variables | symbolic function | symbolic matrix variable | symbolic matrix function | symbolic expression
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
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.
Version History
Introduced in R2013aR2022a: Compute classical adjoint of symbolic matrix functions
The adjoint
function accepts an input argument of type
symfunmatrix
.
R2021a: Compute classical adjoint of symbolic matrix variables
The adjoint
function accepts an input argument of type
symmatrix
.
R2018b: Compute classical adjoint of numeric matrices
The adjoint
function accepts a numeric matrix as an input
argument.
The adjoint
function supports numeric matrices of type
double
and single
, as well as symbolic matrices of
type sym
and symfun
.
See Also
ctranspose
| det
| inv
| rank
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