Adjugate matrix

In linear algebra, the adjugate of a square matrix is a matrix which plays a role similar to the inverse of a matrix, but without the need to perform any divisions.

The adjugate has sometimes been called the "adjoint", but this terminology is ambiguous and is not used in Wikipedia.

Suppose R is a commutative ring and A is an n-by-n matrix over R. The adjugate of A, written as adj(A), is the n-by-n matrix defined by

$\operatorname {adj} (A)[i,j]=(-1)^{i+j}\det(A(j|i))$

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.

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

$A\cdot \operatorname {adj} (A)=\operatorname {adj} (A)\cdot A=\det(A)I_{n}$

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.

The adjucate appears in the formula of the derivative of the determinant.