Exercise 20, § 8

Let $A$ be a commutative ring, $M$ an arbitrary $A$-module and $U = (\alpha_{ij})$ a matrix of type $(m, n)$ with elements in $A$.

(a) Suppose that $m = n$ and let $\Delta = \det(U)$; then, if $x_1, \ldots, x_n$ are elements of $M$ such that $\sum_{j=1}^n \alpha_{ij} x_j = 0$ for $1 \leq i \leq n$, then $\Delta x_i = 0$ for every index $i$.

(b) Suppose that $m$ and $n$ are arbitrary, that all the minors of $U$ of order $> r$ are zero and that a minor of order $r$ is invertible, for example that of the matrix $(\alpha_{ij})$ where $i \leq r$ and $j \leq r$; then if $a_i$ denotes the row of index $i$ in $U$, the $a_i$ of index $\leq r$ form a basis of the submodule of $A^n$ generated by all the rows of $U$; for $i > r$, therefore $a_i = \sum_{k=1}^r \lambda_{ik} a_k$. Then let $y_1, \ldots, y_m$ be elements of $M$; for there to exist elements $x_1, \ldots, x_n$ of $M$ satisfying the system of equations $\sum_{j=1}^n \alpha_{ij} x_j = y_i$ for $1 \leq i \leq m$, it is necessary and sufficient that $y_i = \sum_{k=1}^r \lambda_{ik} y_k$ for $i > r$.