[1][2] It is occasionally known as adjunct matrix,[3][4] or "adjoint",[5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
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.
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 A and B, 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 A or B is not invertible.
A corollary of the previous formula is that, for any non-negative integer k, If A is invertible, then the above formula also holds for negative k. From the identity we deduce Suppose that A commutes with B. Multiplying the identity AB = BA on the left and right by adj(A) proves that If A is invertible, this implies that adj(A) also commutes with B.
Over the real or complex numbers, continuity implies that adj(A) commutes with B even when A 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 2n + 1 elements (e.g. a 5 × 5 matrix over the integers modulo 11).
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).
When A is not invertible, the adjugate satisfies different but closely related formulas.
Collecting these determinants for the different possible i yields an equality of column vectors This formula has the following concrete consequence.
Consider the linear system of equations Assume that A is non-singular.
Multiplying this system on the left by adj(A) and dividing by the determinant yields Applying the previous formula to this situation yields Cramer's rule, where xi is the ith entry of x.
Since p(A) = 0 by the Cayley–Hamilton theorem, some elementary manipulations reveal In particular, the resolvent of A is defined to be and by the above formula, this is equal to The adjugate also appears in Jacobi's formula for the derivative of the determinant.
If A(t) is continuously differentiable, then It follows that the total derivative of the determinant is the transpose of the adjugate: Let pA(t) be the characteristic polynomial of A.
The Cayley–Hamilton theorem states that Separating the constant term and multiplying the equation by adj(A) gives an expression for the adjugate that depends only on A and the coefficients of pA(t).
These coefficients can be explicitly represented in terms of traces of powers of A using complete exponential Bell polynomials.
The resulting formula is where n is the dimension of A, and the sum is taken over s and all sequences of kl ≥ 0 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 A.
The adjugate can be viewed in abstract terms using exterior algebras.
The exterior product defines a bilinear pairing
Pullback by the (n − 1)th exterior power of T induces a morphism of Hom spaces.
Fix a basis vector ei of Rn.
On basis vectors, the (n − 1)st exterior power of T is
If V is endowed with an inner product and a volume form, then the map φ can be decomposed further.
In this case, φ can be understood as the composite of the Hodge star operator and dualization.
Specifically, if ω is the volume form, then it, together with the inner product, determines an isomorphism
A vector v in Rn corresponds to the linear functional
By the definition of the Hodge star operator, this linear functional is dual to *v. That is, ω∨∘ φ equals v ↦ *v∨.
matrix, denoted adjr A, whose entries are indexed by size r subsets I and J of {1, ..., m} [citation needed].
denote the submatrix of A containing those rows and columns whose indices are in Ic and Jc, respectively.
Iteratively taking the adjugate of an invertible matrix A k times yields For example,