I have two square matrices $X = \begin{bmatrix} A&0 \\ C&D \end{bmatrix}$ and $Y = \begin{bmatrix} D&C \\ 0&A \end{bmatrix}$ where $A,D,C$ are rectangular matrices. Could you please tell me how can I prove $det(X)=det(Y)$?
P/S: At first, I thought that I could have applied Schur complements. Unfortunately, these matrices $A,D, C$ are rectangular.
Your statement is almost true. In particular, $Y$ is obtained by permuting the rows and the columns of $X$, i.e., $Y = P X Q$, where $P$ and $Q$ are permutation matrices. Thus, in general $$ \det(Y) = \det(P)\det(X)\det(Q) = \pm \det(X). $$ For example, if we consider $$ X = \left[ \begin{array}{cc|cc} 1 & 0 & 0 & 0 \\\hline 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right], $$ then we have $$ Y = \left[ \begin{array}{cc|cc} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\\hline 0 & 0 & 1 & 0 \\ \end{array} \right]. $$ In this case, $\det(X)=1$, but $\det(Y) = -1$