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?

$$
\begin{bmatrix}
A & B \\
B^T & C
\end{bmatrix}
\begin{bmatrix}
U & V \\
W & X
\end{bmatrix}=
\begin{bmatrix}
AU+BW & AV+BX \\
B^TU+CW & B^TV+CX \\
\end{bmatrix}
$$
So we want


*

*$B^TU+CW=0$, hence $W=-C^{-1}B^TU$

*$AU+BW=I$, so $AU-BC^{-1}B^TU=I$ and therefore $U=(A-BC^{-1}B^T)^{-1}$


Similarly for $V$ and $X$:


*

*$B^TV+CX=I$, so $X=C^{-1}(I-B^TV)$

*$AV+BX=0$, so $AV+BC^{-1}(I-B^TV)=0$, so
$$
AV+BC^{-1}-BC^{-1}BTV=0
$$
and $V=-(A-BC^{-1}B^T)BC^{-1}$


You can compute $X$.
