A equation about determinant I read from the Wiki that: 

When A = D and B = C, the blocks are square matrices of the same order and the following formula holds (even if A and B do not commute)
  $$\det\begin{pmatrix}
A & B\\
B & A
\end{pmatrix}=\det(A-B)\det(A+B).$$

How to prove this relation when $A$ and $B$ do not commute?

Substitute the row $i$ by $R_i-R_{i+n}$ for each $i\in \{1,\dots,n\}$, where $n$ is the dimension of $A$ and $B$. We get 
$$\det\begin{pmatrix}
A & B\\
B & A
\end{pmatrix}=\det\begin{pmatrix}
A-B & B-A\\
B & A
\end{pmatrix}.$$
Now, consider the matrix $\begin{pmatrix}
A-B & B-A\\
B & A
\end{pmatrix}$ and substitute the column $R_{n+j}$ by $R_{n+j}+R_j$ for each $j\in \{1,\dots,n\}$ to get 
$$
\det\begin{pmatrix}
A-B & B-A\\
B & A
\end{pmatrix}=\det\begin{pmatrix}
A-B & 0\\
B & A+B
\end{pmatrix}.$$ 
