Answer
$\det (A)=72$
Work Step by Step
We have the cofactor expansion theorem is:
$\det (A)=a_{14}C_{14}+a_{24}C_{24}+a_{34}C_{34}+a_{44}C_{44}$ with $C_{ij}=(-1)^{i+j}.M_{ij}$ to evaluate the given determinant of column 1:
$\det (A)=0.(-1)^{1+4}C_{14}-2.(-1)^{2+4}C_{24}+4.(-1)^{3+4}C_{34}+0.(-1)^{4+4}C_{44}$
$\det (A)=0-2\begin{vmatrix} 1 & -2 & 3
\\ 0 & 1 & 3 \\
1 & 5 & -2
\end{vmatrix}
-4\begin{vmatrix}1 & -2 & 3
\\ 4 & 0 &7 \\
1 & 5 & -2 \end{vmatrix}+0$
Plug in the given values:
$\det (A)=-2(-17-0-9)-4(-30+25)$
$\det (A)=52+20$
$\det (A)=72$
