Rank of certain matrix determined by blocks

Question

Let $\mathbb F$ be a field and $k,n,r$ be positive integers such that $1\le r \le \min \{k,n\}$. Let $A=\begin{bmatrix}B&C\\D&E\\ \end{bmatrix} \in M_{k\times n}(\mathbb F)$ , where $B \in GL_r(\mathbb F)$ (i.e. $B$ is an invertible $r\times r$ matrix ) .

Then how to show that rank of $A$ is $r$ if and only if $DB^{-1}C=E$ ?

Answer 1

HINT: Suppose $B = I_{r\times r}$. Can you do row operations to get rid of $D$? What must be true about the lower-right $(k-r)\times(n-r)$ block if the rank of $A$ is to be $r$? Then do the general case.

EDIT: In the case I suggested, can you see why row operations give $$\begin{bmatrix} I & C \\ D & E \end{bmatrix}\rightsquigarrow \begin{bmatrix} I & C \\ 0 & E-DC \end{bmatrix}?$$ Thus, you must have $E-DC = 0$ for the matrix to have rank $r$.