I found this identity in [Wikipedia](https://en.wikipedia.org/wiki/Determinant#Block_matrices),

> 
> When $D$ is a $1 \times 1$ matrix, $B$ is a column vector, and $C$ is a row vector, then
> 
> 
> $$\det{\begin{bmatrix} A & B \\ C & D\end{bmatrix}}=(D-1)\det(A) +\det(A-BC)= (D+1)\det(A) -\det(A+BC)$$
> 
> 
> 

I wonder how to prove that. Thanks!