<s>[INST] Prove a matrix has an inverse Prove the matrix
\begin{bmatrix}
A & B \\
B^T & C
\end{bmatrix}
has the inverse
\begin{bmatrix}
D & -DBC^{-1} \\
-C^{-1}B^TD & C^{-1} + C^{-1}B^TDBC^{-1}
\end{bmatrix}
whenever C and D = $(A - BC^{-1}B^T)^{-1}$ are nonsingular.    
I tried to see if their multiplication results in $I_{2}$, but I got stuck.
 [/INST] Note that
\begin{align*}
&\begin{bmatrix}
A & B \\ B^T & C
\end{bmatrix}
\begin{bmatrix}
D & -DBC^{-1} \\ -C^{-1}B^TD & C^{-1}+C^{-1}B^TDBC^{-1}
\end{bmatrix}
\\ &=
\begin{bmatrix}
AD - BC^{-1}B^TD & -ADBC^{-1} + BC^{-1} + BC^{-1}B^TDBC^{-1} \\
B^TD + -CC^{-1}B^TD & -B^TDBC^{-1}+CC^{-1}+CC^{-1}B^TDBC^{-1}
\end{bmatrix}
\\ &=
\begin{bmatrix}
(A-BC^{-1}B^T)D & (I-(A-BC^{-1}B^T)D)BC^{-1} \\
0 & I
\end{bmatrix}
\\ &=
\begin{bmatrix}
I & 0 \\
0 & I
\end{bmatrix}.
\end{align*}
If instead you choose to multiply in the reverse order, then you will have similar cancellations using $D=(A-BC^{-1}B^T)^{-1}$ and the fact that $C$ is invertible.
</s>