Answer
22
Work Step by Step
(a) $|A|=\begin{vmatrix}
-1 & 4 & 1\\
0 & 2 & 2\\
2 & 2 & -3
\end{vmatrix}=(-1).2.(-3)+4.2.2+1.0.2-1.2.2-(-1).2.2-0.4.(-3)=22$
b) Using elementary row operations:
$|A|=\begin{vmatrix}
-1 & 4 & 1\\
0 & 2 & 2\\
2 & 2 & -3
\end{vmatrix} \approx \begin{vmatrix}
-1 & 4 & 1\\
0 & 2 & 2\\
0 & 10 & -1
\end{vmatrix} \approx \begin{vmatrix}
-1 & 4 & 1\\
0 & 2 & 2\\
0 &0 & -11
\end{vmatrix}=22$
c) Using the Cofactor Expansion Theorem
$|A|=A_{11}C_{11}+A_{12}C_{12}+A_{13}C_{13}=-1.(-10)-4.(-4)+1.(-4)=22$
