I was told this is a very basic concept in linear algebra but I am not sure why it is true. Suppose $ [ \mathbf{D}_{\mathbf{x}} F(0), \mathbf{D}_{\mathbf{y}} F(0) ]$ is an $1\times 2$ block matrix with "actual" dimension $m\times n$ if it is viewed not in block form. This matrix has rank $m$.
Why is it, that this fact implies that the $m\times m$ matrix $\mathbf{D}_{\mathbf{y}} F(0)$ is invertible? I thought that this would not imply that $\mathbf{D}_{\mathbf{y}} F(0)$ is invertible because the leading 1s might not all end up in the last $m$ columns.