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 = ( dij )i,j=1,...,n, where dij is the determinant of the matrix Mij and Mij 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).

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