Proposition 16

Let $L = (\lambda_{ij})$ be the matrix (of type $(m, n)$) of system (41). Let $M$ be the matrix of type $(m, n+1)$ obtained by adding to $L$ the $(n+1)$-th column $(\eta_i)$ (II, § 10, no. 1). Let $p$ be the rank of $L$ (calculated by applying the Corollary to Proposition 15). Suppose that the minor $\Delta$ of $L$, the determinant of the matrix by suppressing the rows and columns of index $\geq p + 1$ in $L$, is $\neq 0$ (which is always possible by means of a suitable permutation on the rows of $L$ and a suitable permutation on the columns of $L$). Then, for system (41) to have at least one solution it is necessary and sufficient that all the minors of order $p + 1$ of $M$, the determinants of the submatrices of order $p + 1$ of $M$ whose columns have indices $1, 2, \ldots, p$ and $n+1$, be zero. If this is so, the solutions of system (41) are those of the system consisting of the first $p$ equations; if they are written

all the solutions of this system are obtained by taking for the $\xi_k$ of index $k > p$ arbitrary values and applying Cramer's formulae (no. 7, formulae (37)) to calculate the $\xi_j$ of index $j \leq p$.

We know (II, § 10, no. 12, Proposition 12) that for the system (41) to have at least one solution, it is necessary and sufficient that the matrices $L$ and $M$ have the same rank. With the rows and columns of $L$ permuted to satisfy the condition of the statement, let $a_i$ ($1 \leq i \leq p$) denote the first $p$ columns of $L$ and $y = (\eta_i)$ the $(n+1)$-th column of $M$; since all the columns of $L$ are by hypothesis linear combinations of the $a_i$, to say that $M$ has the same rank $p$ as $L$ means that $y$ is a linear combination of the $a_i$, or also (Proposition 15) that $a_1 \wedge \cdots \wedge a_p \wedge y = 0$. The possibility condition in the statement is the translation of the latter relation, taking account of formula (17) of no. 5. Moreover, since the first $p$ rows of $M$ are linearly independent, the rows of index $> p$ are linear combinations of them and hence every solution of (42) is also a solution of (41). The last assertion is then an immediate consequence of Proposition 13 of no. 7.