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?
Best Answer
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.