Lemma 1

Let $X = (\xi_{ij})$ be a matrix of type $(m, n)$ over a ring $A$. Suppose that $m \geqslant 2$ and that there exists an element $\xi_{i1}$ in the first column of $X$ which is invertible in $A$. Then there exist two invertible square matrices $P \in \mathbf{M}_m(A)$, $Q \in \mathbf{M}_n(A)$ and a matrix $Y$ of type $(m - 1, n - 1)$ over $A$ such that $P$ (resp. $Q$) is a product of matrices of the form $B_{ij}(\lambda)$ of order $m$ (resp. $n$) and

The matrix $B_{ij}(\lambda)X$ is obtained by adding to the row of $X$ of index $i$ the row of index $j$ multiplied on the left by $\lambda$ (no. 9, Example 2); if $\xi_{i1}$ is invertible, then there exists $\lambda \in A$ such that, for the matrix $X' = B_{1i}(\lambda)X = (\xi'_{kl})$, $\xi'_{11} = 1$; multiplying $X'$ on the left by suitably chosen matrices $B_{k1}(\mu_k)$ of order $m$ (for $1 \leq k \leq m$), a matrix $X'' = (\xi''_{kl})$ is obtained such that $\xi''_{j1} = 1$, $\xi''_{k1} = 0$ for $k \neq 1$. Then the matrix obtained is multiplied successively on the right by suitable matrices $B_{1j}(v_j)$ of order $n$ $(2 \leq j \leq n)$ and a matrix is obtained of the form (57).