Nonsingularity and inverse of a matrix. I am having some trouble solving this question: 
Let $A,B,C,D$ be $n\times n$ matrices. Suppose $C$ and $A-BC^{-1}B^T$  are nonsingular.
Show that the matrix mentioned is nonsingular and find its inverse
\begin{bmatrix}A&B\\B^T&C\end{bmatrix}
For nonsingularity $\det(M)$ is not $0$.
Any help is appreciated :)
Edit:
I found that M is nonsingular, can anyone help me finding the inverse?

Suppose $A, B, C$, and $D$ are matrices of dimension $n × n$, $n × m$, $m × n$, and $m × m$, respectively.
In general, when $D$ is invertible, a similar identity with ${\displaystyle \det(D)}$  factored out can be derived:
$${\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(D)\det(A-BD^{-1}C).}$$  
See more on the Wikipedia article here. 
