Proposition 12

For a system of m linear equations in n unknowns

over a field K to have at least one solution, it is necessary and sufficient that the matrix $A = (a_{ij})$ of the system and the matrix B, obtained by bordering A with an $(n+1)$-th column equal to $(b_i)$, be matrices of the same rank.

It has been seen (no. 4) that the existence of a solution of (54) is equivalent to the fact that the column $(b_i)$ is a linear combination of the columns of $A$ and the proposition therefore follows from § 7, no. 3, Corollary 4 to Proposition 4.

Note that the condition of Proposition 12 is always fulfilled when $m = n$ and $A$ is invertible, that is of rank $n$ (Proposition 11). If $x$ and $b$ then denote the matrices with one column $(x_i)$ and $(b_i)$ respectively, system (54) is equivalent to $A.x = b$ and its unique solution is $x = A^{-1}.b$.