Adjugate matrix

In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix. It is also occasionally known as adjunct matrix, though this nomenclature appears to have decreased in usage.
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 formula_6. 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.
Over the complex numbers,
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 "n"×"n" matrix has entries over a field with at least 2"n"+1 elements (e.g. a 5×5 matrix over the integers mod 11). is a polynomial in "t" with degree at most "n", so it has at most n roots. Note that the "ij"th entry of is a polynomial of at most order "n", and likewise for . These two polynomials at the "ij"th entry agree on at least "n"+1 points, as we have at least "n"+1 elements of the field where 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:
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.
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, formula_53 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 formula_55, 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 formula_58 the basis
Fix a basis vector of . The image of under formula_60 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 formula_63 except the term. Therefore, the pullback of formula_63 is the linear transformation for which
that is, it equals
Applying the inverse of formula_60 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 formula_72 matrix, denoted , whose entries are indexed by size subsets and of . Let and denote the complements of and , respectively. Also let formula_73 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:
Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting formula_76 and formula_77 for formula_78 and formula_79, respectively.
Iterated adjugates.
Iteratively taking the adjugate of an invertible matrix A times yields
For example,