# 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 with the (*j*, *i*)'th entry containing

- (-1)
^{i+j} *M*_{ij} = *C*_{ij}

where

*M*_{ij} represents a (

*n*-1)×(

*n*-1)

minor of A, and

*C*_{ij} represents the matrix

cofactor.

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 dividing throughout by 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.