Let A, B, C, D be matrices of order 2 with elements over field F.
Suppose M is block-matrix that in her first row are A and B and in second row C and D.
M is matrix of order 4.
Is det(M) = det(A)*det(D) - det(B)*det(C) ?
I think it's not. Let $\displaystyle A = \begin{pmatrix}1&2\\ 1&1\end{pmatrix}$, $\displaystyle B = \begin{pmatrix}3&4\\ 1&1\end{pmatrix}$, $\displaystyle C = \begin{pmatrix}1&1\\ 0&1\end{pmatrix}$ and $\displaystyle D = \begin{pmatrix}1&1\\ 1&1\end{pmatrix}$.
We have $\displaystyle M=\begin{pmatrix} 1&2&3&4\\
1&1&1&1\\
1&1&1&1\\
0&1&1&1
\end{pmatrix}$ so $\displaystyle \det M = 0$ but
$\displaystyle \det A\cdot \det D -\det C \dot \det D = \left(-1\right)\cdot 0-1\cdot\left(-1\right) = 1 $