Answer
$\det (A)=-52$
Work Step by Step
We have the cofactor expansion theorem is:
$\det (A)=a_{11}C_{11}+a_{21}C_{21}+a_{31}C_{31}+a_{41}C_{41}$ with $C_{ij}=(-1)^{i+j}.M_{ij}$ to evaluate the given determinant of column 1:
$\det (A)=-3.(-1)^{1+1}C_{11}+0.(-1)^{2+1}C_{21}+1.(-1)^{3+1}C_{31}+0.(-1)^{4+1}C_{41}$
$\det (A)=-3\begin{vmatrix} 4 &0 & 2
\\ 4 & -4 & 2 \\
2 & 5 &0
\end{vmatrix}-0\begin{vmatrix} 0 &-1 & 0
\\ 4 & -4 & 2 \\
2 & 5 &0
\end{vmatrix}
+1\begin{vmatrix} 0 & -1 & 0
\\ 4 & 0 & 2 \\
2 & 5 &0
\end{vmatrix}+0.\begin{vmatrix} 0 &-1 & 0
\\ 4 & 0 & 2 \\
4 & -4 &2
\end{vmatrix}$
Plug in the given values:
$\det (A)=-3(-40+56)+1(-4)$
$\det (A)=-48-4$
$\det (A)=-52$
