Corollary 2

For every field $\mathbf{K}$, the group of invertible matrices $\mathbf{GL}(n, \mathbf{K})$ is generated by the permutation matrices (no. 7, Example 2), the diagonal matrices $\operatorname{diag}(a, 1, \ldots, 1)$ ($a \neq 0$ in $\mathbf{K}$) and the matrices $B_{12}(\lambda)$ ($\lambda \in \mathbf{K}$).

It has been seen (no. 9) that the right (resp. left) product of a matrix by the matrix of a suitable transposition exchanges any two columns (resp. rows). Then the matrix $\operatorname{diag}(1, \ldots, 1, a)$ is equal to the product of $\operatorname{diag}(a, 1, \ldots, 1)$ and permutation matrices and every matrix $B_{ij}(\lambda)$ is equal to the product of $B_{12}(\lambda)$ and permutation matrices, whence the corollary.