Exercise 19, § 8

Let $X$ be a matrix over a commutative field. For $X$ to be of rank $p$, it suffices that there exist a minor of $X$ of order $p$ which is $\neq 0$ and such that all the minors of order $p + 1$ containing this minor of order $p$ are zero (show that every column of $X$ is then a linear combination of the $p$ columns to which the elements of the minor of order $p$ in question belong).