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 complex conjugate.

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 defined by

adj(A)[i,j] = (-1)^i+j det(A(j|i))

where A(j|i) denotes the (n-1)-by-(n-1) matrix obtained from A by deleting row j and column i, and det(A(j|i)) is its determinant (the determinant of the 0-by-0 matrix being defined as 1).

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.

We have

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 appears in the formula of the derivative of the determinant.