Corollary

For two matrices over a field, of type $(m, n)$, to be equivalent, it is necessary and sufficient that they have the same rank.

We shall now recover Proposition 13 by another more explicit method. For every ring $A$, every $\lambda \in A$, every integer $m > 1$ and every ordered pair of distinct integers $i, j$ in $[1, m]$, we write

an invertible matrix of order $m$ by no. 8.