Exercise 17, § 8

Let

(*)

be a system of $m$ linear equations in $n$ unknowns over a commutative ring $\mathbf{A}$. Let $x_j$ ($1 \leq j \leq n$) be the columns of the matrix $X = (\alpha_{ij})$ of this system and $y = (\beta_i)$; suppose that in $X$ all the minors of order $> p$ are zero but that $x_1 \wedge x_2 \wedge \cdots \wedge x_p \neq 0$. For the system (*) to have a solution, it is necessary that $x_1 \wedge x_2 \wedge \cdots \wedge x_p \wedge y = 0$. Conversely, if this condition holds, there exist $n + 1$ elements $\xi_j$ ($1 \leq j \leq n + 1$) of $\mathbf{A}$ such that $\xi_{n+1} \neq 0$ and