Answer
136
Work Step by Step
To find the value of the determinant
$\left|\begin{array}{cccc}1 & -2 & 7 & 9 \\ 3 & -4 & 5 & 5 \\ 3 & 6 & 1 & -1 \\ 4 & 5 & 3 & 2\end{array}\right|$
we multiply the first row with (-3) and add it to the second row:
$$
\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
3 & -4 & 5 & 5 \\
3 & 6 & 1 & -1 \\
4 & 5 & 3 & 2
\end{array}\right|=\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
0 & 2 & -16 & -22 \\
3 & 6 & 1 & -1 \\
4 & 5 & 3 & 2
\end{array}\right|
$$
Now, we multiply the first row with ( -3 ) and add it to the third row. Furthermore, we multiply the first row with (-4) and add it to the fourth row:
$$
\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
3 & -4 & 5 & 5 \\
3 & 6 & 1 & -1 \\
4 & 5 & 3 & 2
\end{array}\right|=\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
0 & 2 & -16 & -22 \\
0 & 12 & -20 & -28 \\
0 & 13 & -25 & -34
\end{array}\right|
$$
Taking as the common factor the number $\ 2$ from the second row and the common factor $\ 4$ from the third row, we get
$$
\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
3 & -4 & 5 & 5 \\
3 & 6 & 1 & -1 \\
4 & 5 & 3 & 2
\end{array}\right|=8\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
0 & 1 & -8 & -11 \\
0 & 3 & -5 & -7 \\
0 & 13 & -25 & -34
\end{array}\right|
$$
Multiply the second row with (-3) and add it to the third row. Next, we multiply the second row with (-13) and add it to the fourth row to obtain
$$
\begin{array}{l}
\left|\begin{array}{ccc}
1 & -2 & 7 & 9 \\
3 & -4 & 5 & 5 \\
3 & 6 & 1 & -1 \\
4 & 5 & 3 & 2
\end{array}\right|=8\left|\begin{array}{cccc}
1 & -2 & 7 & 9 \\
0 & 1 & -8 & -11 \\
0 & 0 & 19 & 26 \\
0 & 0 & 79 & 109
\end{array}\right| \\
=8 *(19 * 109-26 * 79)=8 * 17=136
\end{array}
$$
