gives the adjugate of a square matrix m.

# Details • 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)

### Special Matrices(4)

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

The identity matrix is its own adjugate:

## Applications(4)

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: