Exercise 18, § 4

Let C be a principal ideal domain. Consider a system of linear equations

where the coefficients and the right hand sides belong to C, and let A denote the matrix $(\alpha_{ij})$ with m rows and n columns, and B the matrix obtained by extending A by the (n + 1)-st column ($\beta_i$). Show that the system admits at least one solution in $C^n$ if and only if the following conditions are satisfied : 1) A and B have the same rank p ; 2) a gcd of the minors of order p of A is equal to a gcd of the minors of order p of B (use Prop. 4 of VII, p. 20 and Ex. 9, c)).