Corollary

Let $X$ be a matrix of type $(m, n)$ over a commutative field. The rank of $X$ is equal to the greatest integer $p$ such that there exists at least one minor of $X$ of order $p$ which is $\neq 0$.

The rank of $X$ is the maximum number of linearly independent columns of $X$ (II, § 10, no. 12, Definition 7). The corollary then follows from Proposition 15 and formula (17) of no. 5.

Consider now the case of a system of $m$ linear equations in $n$ unknowns over a commutative field $\mathbf{K}$:

(41)