Adjugate

In linear algebra, the adjugate of a square matrix is a matrix which plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions.

The adjugate has sometimes been called the "adjoint", but that terminology is ambiguous and is not used in Wikipedia. Today, "adjoint" normally refers to the conjugate transpose.

Definition

Suppose R is a commutative ring and A is an n-by-n matrix with entries from R. The adjugate of A, written as adj(A), is the n-by-n matrix whose entry in row j and column i is given by

(-1)^i+j M_ij = C_ij

where M_ij represents the (n-1)×(n-1) minor of A (obtained by deleting row i and column j of A and taking the determinant of the resulting (n-1)-by(n-1) matrix), and C_ij represents the matrix cofactors.

Example

As an example, we have

Here, the -4 in the last row, second column was computed by deleting the second row and last column of the original matrix and computing

Properties

As a consequence of Laplace's formula for the computation of determinants, we have

A adj(A) = adj(A) A = det(A) I_n

where I_n denotes the n-by-n identity matrix. This formula is used to prove that A is invertible as a matrix over R if and only if det(A) is invertible as an element of R. As another consequence of this, we can find the inverse easily from the adjugate - multiplying adjugate by the inverse of the determinant yields the identity matrix. From the above equation this is clear by multiplying throughout by A^-1/det(A):

A^-1 = adj(A) / det(A).

We have the properties

adj(I_n) = I_n

and

adj(AB) = adj(B) adj(A)

for all n-by-n matrices A and B. The adjugate is also compatible with transposition:

adj(A^T) = (adj(A))^T.

Furthermore,

det(adj(A)) = det(A)^n-1.

If p(t) = det(A - tI_n) is the characteristic polynomial of A and we define the polynomial q(t) = (p(0) - p(t))/t, then

adj(A) = q(A).

The adjugate also appears in the formula of the derivative of the determinant.