Proposition 11

Let X be a square matrix of order n over a field K. The following properties are equivalent:

(a) X is invertible in $\mathbf{M}_n(\mathbf{K})$. (b) X is right invertible in $\mathbf{M}_n(\mathbf{K})$. (c) X is left invertible in $\mathbf{M}_n(\mathbf{K})$. (d) X is of rank n.

This is just a translation of § 7, no. 4, Corollary to Proposition 9.