The comatrix com(M) of a n x n-matrix M is the n x n-matrix whose entry in (l,k) is \small{(-1)^{l+k}} times the determinant of the matrix which you get by deleting in M the line l and the row k.
It is the transpose of the comatrix which is of interest to us; it is called adjugate matrix, and it is shown in every lecture on linear algebra that it has the following fundamental property:

^t\text{com}(M)\:M\;=\;M\:^t\text{com}(M)\;=\;\det(M)\:I\:.

If you work with matrix coefficients in a ring R, it follows that the matrix M is invertible in the matrix ring precisely if the scalar det(M) is invertible in the ring R. For example over \small\mathbb{Z} the invertible matrices are those whose determinant is 1 oder -1 ist.

Exercise:  Prove that  com  preserves matrix multiplication, i.e.,

com(I) = I      et      com(MN) = com(M) com(N).