Employ Theorem CINM,
\begin{equation*}
\begin{bmatrix}
1 & 1 & 3 & 1 & 1 & 0 & 0 & 0\\
-2 & -1 & -4 & -1 & 0 & 1 & 0 & 0\\
1 & 4 & 10 & 2 & 0 & 0 & 1 & 0\\
-2 & 0 & -4 & 5 & 0 & 0 & 0 & 1
\end{bmatrix}
\rref
\begin{bmatrix}
\leading{1} & 0 & 0 & 0 & 38 & 18 & -5 & -2\\
0 & \leading{1} & 0 & 0 & 96 & 47 & -12 & -5\\
0 & 0 & \leading{1} & 0 & -39 & -19 & 5 & 2\\
0 & 0 & 0 & \leading{1} & -16 & -8 & 2 & 1
\end{bmatrix}
\end{equation*}
And therefore we see that $C$ is nonsingular ($C$ row-reduces to the identity matrix, Theorem NMRRI) and by Theorem CINM,
\begin{equation*}
\inverse{C}=
\begin{bmatrix}
38 & 18 & -5 & -2\\
96 & 47 & -12 & -5\\
-39 & -19 & 5 & 2\\
-16 & -8 & 2 & 1
\end{bmatrix}
\end{equation*}