Proposition 6

In the ring $\mathbf{M}_n(\mathbf{R})$, the group $\mathbf{GL}_n(\mathbf{R})$ of non-singular matrices is a dense open subset, and the topology induced on this set is compatible with its group structure.

(*) If $E, F, G$ are three vector spaces over a field $K$, a mapping $f$ of $E \times F$ into $G$ is said to be bilinear if we have identically

for all $x, x' \in E$, all $y, y' \in F$ and all $\lambda \in K$.

If $X$ is a nonsingular square matrix, the elements of $X^{-1}$ are rational functions of the elements of $X$; these functions are therefore defined and continuous in a neighbourhood of $X$, so that every matrix $Y$ in this neighbourhood is non-singular, and the mapping $Y \to Y^{-1}$ is continuous at the point $X$; hence $\mathbf{GL}_n(\mathbf{R})$ is open in $\mathbf{M}_n(\mathbf{R})$ and the topology of $\mathbf{GL}_n(\mathbf{R})$ is compatible with its group structure.

Finally, $\mathbf{GL}_n(\mathbf{R})$ is the complement of the set of square matrices $X$ whose determinant is zero; since the determinant of $X$ is a polynomial in the elements of $X$, Proposition 3 of no. 4 shows that $\mathbf{GL}_n(\mathbf{R})$ is dense in $\mathbf{M}_n(\mathbf{R})$.