Answer
24
Work Step by Step
(a) $|A|=\begin{vmatrix}
3 & -1 & -2 & 1\\
0 & 0 &1& 4\\
0 & 2 & 1 & -1\\
0 &0 & 0 & -4
\end{vmatrix}=-3.1.2.(-4)=24$
b) Using elementary row operations:
$|A|=\begin{vmatrix}
3 & -1 & -2 & 1\\
0 & 0 &1& 4\\
0 & 2 & 1 & -1\\
0 &0 & 0 & -4
\end{vmatrix} \approx \begin{vmatrix}
3 & -1 & -2 & 1\\
0 & 2 & 1 & -1 \\
0 & 0 &1& 4\\
0 &0 & 0 & -4
\end{vmatrix} =-3.2.1.(-4)=24$
c) Using the Cofactor Expansion Theorem
$|A|=3\begin{vmatrix}
0 &1& 4\\
2 & 1 & -1 \\
0 &0 & -4
\end{vmatrix}=3.8=24$
