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).