Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$ Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that:
$$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$
under the following assumptions:


*

*$m$ can be any number we want, but I prefer $2n$.

*$E$ should not contain the terms $\det{(A)},\det{(B)},\det{(C)},\det{(D)}$. that means that the matrix 
$\begin{pmatrix}
\det{(A)} & \det{(B)} \\
\det{(C)} & \det{(D)}
\end{pmatrix}
$ is not the case...

*There can't be any further assumptions on $A,B,C,D$


I've already checked the matrix 
$\begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
$
but it's not that...
Does anyone have an idea what $E$ can be?
 A: Note: This answer is wrong, as indicated by the comments below.

Let $M \oplus N$ denote the block-diagonal matrix 
$$
M \oplus N = \pmatrix{M&0\\0&N}
$$
Then one solution with $m = 4n$ is
$$
E = \pmatrix{A \oplus I & I \oplus B\\I \oplus C & D \oplus I}
$$
where $I$ denotes the $n \times n$ identity matrix.
A: Let $s$ be the product of the diagonal entries of $A$ and the diagonal entries of $D$, let $t$ be the product of the diagonal entries of $B$ and the diagonal entries of $C$ and consider 
the $m\times m$ matrix $E=\begin{pmatrix}\det{(A)}\det{(D)}-\det{(B)}\det{(C)}-s+t&s-t\\-1&1\\&&1\\&&&\ddots\\&&&&1\end{pmatrix}$. 
Notice that you can pick any $m$, as long as it is at least $2$.
A: When $m=1$, clearly what you ask for is impossible in general.
When $m\ge2$, let $a=\det(A), b=\det(B), c=\det(C), d=\det(D)$ and $e=\det(AD)-\det(BC)$. There are two cases:


*

*If $e=0$, take $E$ to be $x$ times the all-one matrix for any sufficiently large $x>0$.

*If $e\ne0$, let $S_m$ be the symmetric Pascal matrix of size $m$, which is a positive integer matrix of determinant 1 and whose first row is a row of ones. Now, any sufficiently large $x>0$, we have $x>a,b,c,d$ and $\frac{e}{x^{m-1}}\ne a,b,c,d$. Therefore we may take
$$
E=\operatorname{diag}\left(\frac{e}{x^{m-1}},x,x,\ldots,x\right)\,S.
$$

