Let be R a

commutative ring with 1 and M a n x n

matrix over R.

The

**adjoint matrix of M, adj M** is defined as:

adj M = ( d

_{ij} )

_{i,j=1,...,n}, where d

_{ij} is the

determinant of the matrix M

_{ij} and M

_{ij} is the matrix you get when you delete the i th row and the j th column (a (n-1) x (n-1) matrix !).

Note that the determinant can be defined for matrices over any commutative ring.

The following law holds:

**(det M) I = M adj(M) = adj(M) M**, where I is the unit matrix (diagonal matrix with 1 on the diagonal).