Answer
7441
Work Step by Step
To find the value of the determinant
$$
\left|\begin{array}{cccc}
0 & -3 & 8 & 2 \\
8 & 1 & -1 & 6 \\
-4 & 6 & 0 & 9 \\
-7 & 0 & 0 & 14
\end{array}\right|
$$
we can replace the fourth row with the first row; then we have
$$
\left|\begin{array}{cccc}
0 & -3 & 8 & 2 \\
8 & 1 & -1 & 6 \\
-4 & 6 & 0 & 9 \\
-7 & 0 & 0 & 14
\end{array}\right|=-\left|\begin{array}{cccc}
-7 & 0 & 0 & 14 \\
8 & 1 & -1 & 6 \\
-4 & 6 & 0 & 9 \\
0 & -3 & 8 & 2
\end{array}\right|=7\left|\begin{array}{cccc}
1 & 0 & 0 & -2 \\
8 & 1 & -1 & 6 \\
-4 & 6 & 0 & 9 \\
0 & -3 & 8 & 2
\end{array}\right|
$$
Multiply the first row with (-8) and then add the output to the second row. Furthermore, we multiply the first row with (4) and add the output to the third row:
$$
\left|\begin{array}{cccc}
0 & -3 & 8 & 2 \\
8 & 1 & -1 & 6 \\
-4 & 6 & 0 & 9 \\
-7 & 0 & 0 & 14
\end{array}\right|=7\left|\begin{array}{cccc}
1 & 0 & 0 & -2 \\
0 & 1 & -1 & 22 \\
0 & 6 & 0 & 1 \\
0 & -3 & 8 & 2
\end{array}\right|
$$
Multiply the second row with (-6) and add the output to the third row. Furthermore, we multiply the second row with ( 3 ) and then add the output to the fourth row
$$
\begin{array}{llll}
\left|\begin{array}{ccc}
0 & -3 & 8 & 2 \\
8 & 1 & -1 & 6 \\
-4 & 6 & 0 & 9 \\
-7 & 0 & 0 & 14
\end{array}\right|=7 \left|\begin{array}{ccccc}
1 & 0 & 0 & -2 \\
0 & 1 & -1 & 22 \\
0 & 0 & 6 & -131 \\
0 & 0 & 5 & 68
\end{array}\right| \\
& \cr =7*1*1*[6 * 68+5 * 131]=7441
\end{array}
$$
