Adjugate matrix
In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.

The adjugate has sometimes been called the "adjoint", but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

Definition 
The adjugate of  is the transpose of the cofactor matrix  of ,

In more detail,  suppose  is a commutative ring and  is an  matrix with entries from .  The -minor of , denoted , is the determinant of the  matrix that results from deleting row  and column  of .  The cofactor matrix of  is the  matrix  whose  entry is the  cofactor of , which is the -minor times a sign factor:

The adjugate of  is the transpose of , that is, the  matrix whose  entry is the  cofactor of ,

The adjugate is defined as it is so that the product of  with its adjugate yields a diagonal matrix whose diagonal entries are the determinant .  That is,

where  is the  identity matrix.  This is a consequence of the Laplace expansion of the determinant.

The above formula implies one of the fundamental results in matrix algebra, that  is invertible if and only if  is an invertible element of .  When this holds, the equation above yields

Examples

1 × 1 generic matrix 
The adjugate of any non-zero 1×1 matrix (complex scalar) is .  By convention, adj(0) = 0.

2 × 2 generic matrix 
The adjugate of the 2×2 matrix

is

By direct computation,

In this case, it is also true that det(adj(A)) = det(A) and hence that adj(adj(A)) = A.

3 × 3 generic matrix 
Consider a 3×3 matrix

Its cofactor matrix is

where
.

Its adjugate is the transpose of its cofactor matrix,
.

3 × 3 numeric matrix 
As a specific example, we have

It is easy to check the adjugate is the inverse times the determinant, .

The  in the second row, third column of the adjugate was computed as follows.  The (2,3) entry of the adjugate is the (3,2) cofactor of A.  This cofactor is computed using the submatrix obtained by deleting the third row and second column of the original matrix A,

The (3,2) cofactor is a sign times the determinant of this submatrix:

and this is the (2,3) entry of the adjugate.

Properties 
For any  matrix , elementary computations show that adjugates enjoy the following properties.
  and , where  and  are the zero and identity matrices, respectively.
  for any scalar .
 .
 .
 If  is invertible, then .  It follows that:
  is invertible with inverse .
 .
  is entrywise polynomial in .  In particular, over the real or complex numbers, the adjugate is a smooth function of the entries of .

Over the complex numbers,
 , where the bar denotes complex conjugation.
 , where the asterisk denotes conjugate transpose.

Suppose that  is another  matrix.  Then

This can be proved in three ways.  One way, valid for any 
commutative ring, is a direct computation using the 
Cauchy–Binet formula.  The second way, valid for the real 
or complex numbers, is to first observe that for invertible 
matrices  and ,

Because every non-invertible matrix is the limit of invertible matrices, continuity of the adjugate then implies that the formula remains true when one of  or  is not invertible.

A corollary of the previous formula is that, for any non-negative integer ,

If  is invertible, then the above formula also holds for negative .

From the identity

we deduce

Suppose that  commutes with .  Multiplying the identity  on the left and right by  proves that

If  is invertible, this implies that  also commutes with .  Over the real or complex numbers, continuity implies that  commutes with  even when  is not invertible.

Finally, there is a more general proof than the second proof, which only requires that an nxn matrix has entries over a field with at least 2n+1 elements (e.g. a 5x5 matrix over the integers mod 11). det(A+tI) is a polynomial in t with degree at most n, so it has at most n roots. Note that the ijth entry of adj((A+tI)(B)) is a polynomial of at most order n, and likewise for adj(A+tI)adj(B). These two polynomials at the ijth entry agree on at least n+1 points, as we have at least n+1 elements of the field where A+tI is invertible, and we have proven the identity for invertible matrices. Polynomials of degree n which agree on n+1 points must be identical (subtract them from each other and you have n+1 roots for a polynomial of degree at most n - a contradiction unless their difference is identically zero). As the two polynomials are identical, they take the same value for every value of t. Thus, they take the same value when t = 0. 

Using the above properties and other elementary computations, it is straightforward to show that if  has one of the following properties, then  does as well:
 Upper triangular,
 Lower triangular,
 Diagonal,
 Orthogonal,
 Unitary,
 Symmetric,
 Hermitian,
 Skew-symmetric,
 Skew-hermitian,
 Normal.

If  is invertible, then, as noted above, there is a formula for  in terms of the determinant and inverse of .  When  is not invertible, the adjugate satisfies different but closely related formulas.
 If , then .
 If , then .  (Some minor is non-zero, so  is non-zero and hence has rank at least one; the identity  implies that the dimension of the nullspace of  is at least , so its rank is at most one.)  It follows that , where  is a scalar and  and  are vectors such that  and .

Column substitution and Cramer's rule 

Partition  into column vectors:

Let  be a column vector of size .  Fix  and consider the matrix formed by replacing column  of  by :

Laplace expand the determinant of this matrix along column .  The result is entry  of the product .  Collecting these determinants for the different possible  yields an equality of column vectors

This formula has the following concrete consequence.  Consider the linear system of equations

Assume that  is non-singular.  Multiplying this system on the left by  and dividing by the determinant yields

Applying the previous formula to this situation yields Cramer's rule,

where  is the th entry of .

Characteristic polynomial 
Let the characteristic polynomial of  be

The first divided difference of  is a symmetric polynomial of degree ,

Multiply  by its adjugate.  Since  by the Cayley–Hamilton theorem, some elementary manipulations reveal

In particular, the resolvent of  is defined to be

and by the above formula, this is equal to

Jacobi's formula 

The adjugate also appears in Jacobi's formula for the derivative of the determinant.  If  is continuously differentiable, then

It follows that the total derivative of the determinant is the transpose of the adjugate:

Cayley–Hamilton formula 

Let  be the characteristic polynomial of .  The Cayley–Hamilton theorem states that

Separating the constant term and multiplying the equation by  gives an expression for the adjugate that depends only on  and the coefficients of .  These coefficients can be explicitly represented in terms of traces of powers of  using complete exponential Bell polynomials.  The resulting formula is

where  is the dimension of , and the sum is taken over  and all sequences of  satisfying the linear Diophantine equation

For the 2×2 case, this gives

For the 3×3 case, this gives

For the 4×4 case, this gives

The same formula follows directly from the terminating step of the Faddeev–LeVerrier algorithm, which efficiently determines the characteristic polynomial of .

Relation to exterior algebras 
The adjugate can be viewed in abstract terms using exterior algebras.  Let  be an -dimensional vector space.  The exterior product defines a bilinear pairing

Abstractly,  is isomorphic to , and under any such isomorphism the exterior product is a perfect pairing.  Therefore, it yields an isomorphism

Explicitly, this pairing sends  to , where

Suppose that  is a linear transformation.  Pullback by the st exterior power of  induces a morphism of  spaces.  The adjugate of  is the composite

If  is endowed with its coordinate basis , and if the matrix of  in this basis is , then the adjugate of  is the adjugate of .  To see why, give  the basis

Fix a basis vector  of .  The image of  under  is determined by where it sends basis vectors:

On basis vectors, the st exterior power of  is

Each of these terms maps to zero under  except the  term.  Therefore, the pullback of  is the linear transformation for which

that is, it equals

Applying the inverse of  shows that the adjugate of  is the linear transformation for which

Consequently, its matrix representation is the adjugate of .

If  is endowed with an inner product and a volume form, then the map  can be decomposed further.  In this case,  can be understood as the composite of the Hodge star operator and dualization.  Specifically, if  is the volume form, then it, together with the inner product, determines an isomorphism

This induces an isomorphism

A vector  in  corresponds to the linear functional

By the definition of the Hodge star operator, this linear functional is dual to .  That is,  equals .

Higher adjugates 
Let  be an  matrix, and fix .  The th higher adjugate of  is an  matrix, denoted , whose entries are indexed by size  subsets  and  of .  Let  and  denote the complements of  and , respectively.  Also let  denote the submatrix of  containing those rows and columns whose indices are in  and , respectively.  Then the  entry of  is

where  and  are the sum of the elements of  and , respectively.

Basic properties of higher adjugates include:
 .
 .
 .
 .
 , where  denotes the th compound matrix.

Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting  and  for  and , respectively.

Iterated adjugates 
Iteratively taking the adjugate of an invertible matrix A   times yields

For example,