TITLE: What am I doing wrong when trying to find a determinant of this 4x4
QUESTION [4 upvotes]: 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:

REPLY [1 votes]: Define the   permutation matrices $P_k$ by $P_1 e_2 = e_3, P_1 e_3 = e_2$, and  $P_2 e_1 = e_3, P_2 e_2 = e_1, P_2 e_3 = e_2$ (the other basis vectors mapped to themselves). It is easy to verify that $\det P_1 = -1, \det P_2 = +1$.
Then $P_1 A P_2 = \begin{bmatrix} 2 & 5 & -7 & 2 \\
0 & -5 & -8 & 3 \\
0 & 0 & 3 & -4 \\
0 & 0 & 5 & -6\end{bmatrix}$, and $\det (P_1 A P_2) = -20$, hence $\det A = 20$.
{"set_name": "stack_exchange", "score": 4, "question_id": 520755}