AdjugateIn 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 Mij = Cij
As a consequence of Laplace's formula for the computation of determinants, we have
- A · adj(A) = adj(A) · A = det(A) In
We have the properties
- adj(In) = In
- adj(AB) = adj(B) adj(A)
- adj(AT) = (adj(A))T.
- det(adj(A)) = det(A)n-1.
- adj(A) = q(A).