What am I doing wrong when trying to find a determinant of this 4x4 I have to find the determinant of this 4x4 matrix:
$
\begin{bmatrix}
 5 & -7 & 2 & 2 \\
 0 & 3 & 0 & -4 \\
 -5 & -8 & 0 & 3 \\
 0 & -5 & 0 & -6 \\
\end{bmatrix}
$

Here is my working which seems wrong according to the solutions. What am i doing wrong?:



And here is the solution:


What you are doing wrong is precisely what the solution said you were doing wrong. The $2$ was alright, since that's the same as $2\cdot(-1)^{1+3},$ but the $-5$ was not, since $$-5\cdot(-1)^{2+1}=-5\cdot-1=5.$$ Keep in mind that we have an alternating sign factor as we move along a row/column, and that the starting sign depends on the row/column that we're in.
