Answer
$\det(A)=31$
Work Step by Step
$\det(A)=-2\begin{vmatrix} -2 & 0 & 1 & 1
\\ 1&2 &2&0 \\
-4 & 4 & 6 &1 \\
-1 & 1 & 0 & 5
\end{vmatrix}=-2\begin{vmatrix}1 &0 &\frac{-1}{2}&\frac{-1}{2}
\\ 1 &2 &2 &0 \\
-4 &4 &6&1 \\
-1 & 1 & 0 & 5
\end{vmatrix}=-2\begin{vmatrix}1 &0 &\frac{-1}{2}&\frac{-1}{2}
\\ 0 &2 &\frac{5}{2} &\frac{1}{2} \\
0 &4 &4&-1 \\
0 & 1 & \frac{-1}{2} &\frac{9}{2}
\end{vmatrix}=-4\begin{vmatrix}1 &0 &\frac{-1}{2}&\frac{-1}{2}
\\ 0 &1 &\frac{5}{4} &\frac{1}{4} \\
0 &4 &4&-1 \\
0 & 1 & \frac{-1}{2} &\frac{9}{2}
\end{vmatrix}=-4\begin{vmatrix}1 &0 &\frac{-1}{2}&\frac{-1}{2}
\\ 0 &1 &\frac{5}{4} &\frac{1}{4} \\
0 &0 &-1&-2 \\
0 & 0 & \frac{-7}{4} &\frac{17}{4}
\end{vmatrix}=4\begin{vmatrix}1 &0 &\frac{-1}{2}&\frac{-1}{2}
\\ 0 &1 &\frac{5}{4} &\frac{1}{4} \\
0 &0 &1&2 \\
0 & 0 & \frac{-7}{4} &\frac{17}{4}
\end{vmatrix}=4\begin{vmatrix}1 &0 &\frac{-1}{2}&\frac{-1}{2}
\\ 0 &1 &\frac{5}{4} &\frac{1}{4} \\
0 &0 &1&2 \\
0 & 0 & 0 &\frac{31}{4}
\end{vmatrix}$
We will apply Corollary3.2.6. The determinant of the matrix of coefficients of the system is:
$\det(A)=a_{11}C_{11}+a_{21}C_{21}+a_{31}C_{31}$
where $C_{ij}=(_1)^{i+j}.M_{ij}$
$\det (A)=4.[1(-1)^{1+1}.1.\frac{31}{4}]$
$\det (A)=4.\frac{31}{4}$
$\det(A)=31$
