Answer
The value of $x$ is, $=4$
Work Step by Step
Consider, $\left| \begin{matrix}
1 & x & -2 \\
3 & 1 & 1 \\
0 & -2 & 2 \\
\end{matrix} \right|=-8$
Evaluate the above determinant as follows:
$\left| \begin{matrix}
1 & x & -2 \\
3 & 1 & 1 \\
0 & -2 & 2 \\
\end{matrix} \right|=-8$
Evaluate the above determinant by expanding along the first column as follows:
$1\left| \begin{matrix}
1 & 1 \\
-2 & 2 \\
\end{matrix} \right|-3\left| \begin{matrix}
x & -2 \\
-2 & 2 \\
\end{matrix} \right|+0=-8$
Next, calculate the above determinant as given:
$\begin{align}
& 2-\left( -2 \right)-3\left( 2x-4 \right)=-8 \\
& 4-6x+12=-8 \\
& -6x+16=-8
\end{align}$
Add $-16$ to both sides to get,
$-6x=-24$
Divide by $-6$. Now,
$x=4$.
Thus, $x=4$
