Answer
$-1100$
Work Step by Step
To find the value of the determinant, we can multiply the first row with (-2) and then add the output to the second row. Furthermore, we multiply the first row with (-2) and then we add the output to the third row. After that, we replace the second row with the fifth row
$$
\left|\begin{array}{ccccc}
1 & -1 & 8 & 4 & 2 \\
2 & 6 & 0 & -4 & 3 \\
2 & 0 & 2 & 6 & 2 \\
0 & 2 & 8 & 0&0 \\
0 & 1 & 1 & 2 & 2
\end{array}\right|=\left|\begin{array}{ccccc}
-1 & -1 & 8 & 4 & 2 \\
0 & 1 & 1 & 2 & 2 \\
0 & 2 &-14 & -2 & -2 \\
0 & 2 & 8 & 0 & 0 \\
0 & 8 & -16 & -12 & -1
\end{array}\right|
$$
Multiply the second row with (-2) and then add the output to the third row. Next, we multiply the second row with (-2) and add the output to the fourth row. Then, we multiply the second row with (-8) and then add the output to the third row. Finally, we replace the third row with the fourth row
$$
\left|\begin{array}{ccccc}
1 & -1 & 8 & 4 & 2 \\
2 & 6 & 0 & -4 & 3 \\
0 & 0 & 2 & 6 & 2 \\
0 & 2 & 8 & 0 & 0 \\
0 & 1 & 1 & 2 & 2
\end{array}\right|=\left|\begin{array}{cccc}
1 & -1 & 8 & 4 & 2 \\
0 & 1 & 1 & 2 & 2 \\
0 & 0 & 6 & -4 & -4 \\
0 & 0 & -16 & -6 & -6 \\
0 & 0 & -24 & -28 & -17 \\
\end{array}\right|
$$
We can multiply the third row with (4) and then add the output to the fifth row. Next, we multiply the third row with $(16 / 6)$ and then we add the output to the fourth row
$$
\left|\begin{array}{cccc}
1 & -1 & 8 & 4 & 2 \\
2 & 6 & 0 & -4 & 3 \\
2 & 0 & 2 & 6 & 2 \\
0 & 2 & 8 & 0&0 \\
0 & 1 & 1 & 2 & 2 \\
\end{array}\right|=\left|\begin{array}{cccc}
1 & -1 & 8 & 4 & 2 \\
0 & 1 & 1 & 2 & 2 \\
0 & 0 & 6 & -4 & -4 \\
0 & 0 & 0 & -50/3 & -50/3 \\
0 & 0 & 0 & -44 & -33 \\
\end{array}\right|\\=6 * (33 * \frac {50}{3}-44 * \frac {50}{3})=-1100
$$
