# Adjugate

Adjugate[m]

gives the adjugate of a square matrix m.

# Details

• The adjugate is also known as the classical adjoint or the adjunct matrix.
• The adjugate of an invertible matrix m is given by Inverse[m]Det[m].
• The matrix product of a matrix m with its adjugate is equal to the determinant of m multiplied by an identity matrix of the same size as m.
• The matrix m can be numerical or symbolic, but must be square.

# Examples

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## Basic Examples(3)

Compute the adjugate of a 2×2 matrix:

Compute the adjugate for a symbolic matrix:

Compute the adjugate for a 3×3 matrix:

Verify the relation with m:

## Scope(9)

### Basic Uses(5)

Adjugate for a machine-precision matrix:

Adjugate for a complex matrix:

Adjugate for an exact matrix:

Adjugate for an arbitrary-precision matrix:

Adjugate for a symbolic matrix:

### Special Matrices(4)

Adjugate of a sparse matrix is returned as a normal matrix:

Adjugates of structured matrices:

The identity matrix is its own adjugate:

Adjugate of a Hilbert matrix:

## Applications(4)

Compute a cofactor using Adjugate:

Verify the function cofactor:

Compute the inverse of a matrix using Adjugate:

Compare with Inverse:

Use Adjugate to solve a linear equation:

Compare with LinearSolve:

Define a function for computing the Gaussian curvature of a surface represented as an implicit Cartesian equation:

The implicit Cartesian equation of a surface with icosahedral symmetry:

Compute its Gaussian curvature:

Visualize regions of positive (red) and negative (blue) Gaussian curvature on the surface:

## Properties & Relations(5)

m.Adjugate[m] is equal to Det[m] times an identity matrix of the same size:

Inverse[m] is equal to the adjugate divided by the determinant:

For an n×n matrix m, Adjugate[m] equals LinearSolve[m,Det[m]IdentityMatrix[n]]:

For an n×n matrix m, Det[Adjugate[m]]==Det[m]n-1:

Minors[m] can be computed using Adjugate:

## Possible Issues(1)

The adjugate is defined only for square matrices:

## Neat Examples(2)

Define a function for computing the adjugate polynomial of a square matrix:

Compute the adjugate polynomial of a matrix:

Evaluating the adjugate polynomial of a matrix at the matrix itself gives the adjugate:

Define a function for computing the iterated adjugate of a square matrix:

Compute the first few iterates for a matrix:

This is equivalent to using NestList with Adjugate:

Wolfram Research (2021), Adjugate, Wolfram Language function, https://reference.wolfram.com/language/ref/Adjugate.html.

#### Text

Wolfram Research (2021), Adjugate, Wolfram Language function, https://reference.wolfram.com/language/ref/Adjugate.html.

#### CMS

Wolfram Language. 2021. "Adjugate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Adjugate.html.

#### APA

Wolfram Language. (2021). Adjugate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Adjugate.html

#### BibTeX

@misc{reference.wolfram_2024_adjugate, author="Wolfram Research", title="{Adjugate}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Adjugate.html}", note=[Accessed: 16-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_adjugate, organization={Wolfram Research}, title={Adjugate}, year={2021}, url={https://reference.wolfram.com/language/ref/Adjugate.html}, note=[Accessed: 16-July-2024 ]}