Answer
-30
Work Step by Step
(a) $|A|=\begin{vmatrix}
0 & 0 & 0 & -2\\
0 & 0 & -5 & 1\\
0 & 1 & -4 & 1\\
-3 & -3 & -3 & -3
\end{vmatrix}=-2.(-5).1(-3)=-30$
b) Using elementary row operations:
$|A|=\begin{vmatrix}
0 & 0 & 0 & -2\\
0 & 0 & -5 & 1\\
0 & 1 & -4 & 1\\
-3 & -3 & -3 & -3
\end{vmatrix} \approx \begin{vmatrix}
-3 & -3 & -3 & -3\\
0 & 0 & -5 & 1\\
0 & 1 & -4 & 1\\
0 & 0 & 0 & -2
\end{vmatrix} \approx \begin{vmatrix}
-3 & -3 & -3 & -3\\
0 & 1 & -4 & 1\\
0 & 0 & -5 & 1\\
0 & 0 & 0 & -2
\end{vmatrix}=-3.1.(-5).(-2)=-30$
c) Using the Cofactor Expansion Theorem
$|A|=A_{11}C_{11}+A_{21}C_{21}+A_{23}C_{23}=-30$
