If an n x n matrix A has a nonzero determinant, then there is a matrix called "A inverse", denoted by A^-1 such that AA^-1 = A^-1A = Id. Here Id denotes the identity matrix, that is the matrix that has ones on its diagonal entries and zeros everywhere else. If there is matrix B, say, that has the property that BA = AB = Id then B = A^-1. Also, the determinant of A is nonzero because the determinant of a product of matrices is equal to the product of the matrices' determinants. Hence det(A)det(B) = det (Id) = 1. So, det(A) = 1/det(A^-1).

The entries of A^-1 can be found with Cramer's rule. Since AA^-1 = Id, if the jth column of A^-1 is denoted by A^-1_j then the jth column of Id is equal to AA^-...