## Syntax

``X = adjoint(A)``

## 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 = 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 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

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

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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.

## Version History

Introduced in R2013a