Permuting the rows of an invertible matrix to make 2 specific submatrices invertible This question is similar to a previous one: Gauss Elimination with constraints

Given an $n \times n$ matrix $M$ and a number $1 \leq m \leq n-1$, we partition is as a block matrix:
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$
where $A$ is an $m \times m$ matrix and $D$ is an $(n-m) \times (n-m)$ matrix. We then say that $M$ is $m$-good if both $A$ and $D$ are invertible.
Given any invertible matrix $M \in GL_n(\mathbb{F})$ and a number $1 \leq m \leq n-1$, is it always possible to permute the rows of $M$ to make it $m$-good?
Note: I only care about the case $\mathbb{F}=\mathbb{Z}_p$, but I asked the question more generally because my feeling is that it doesn't matter what the field is.

Using the Laplace expansion, $\det M$ is a sum of terms of the form $ \pm \det(A) \det(D)$ over all ways of partitioning the rows (see e.g. http://accessscience.com/content/Determinant/188900 : the term "Laplace expansion" is sometimes used for the cofactor expansion along a single row or column, but it's really more general).  So if $\det(A) \det(D)$ was always 0, $\det(M)$ would also be 0. 
