<s>[INST] Can we interchange columns in a determinant? Can we interchange columns in a determinant like this and preserve the value of it? For example:
$$
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 \\ 
2 & 1 & 1 & 1 & 0 \\
3 & 1 & 1 & 0 & 0 \\
4 & 1 & 0 & 0 & 0 \\
5 & 0 & 0 & 0 & 0
\end{pmatrix}
$$
to 
$$
\begin{pmatrix}
1 & 1 & 1 & 1 & 1 \\ 
0 & 1 & 1 & 1 & 2 \\
0 & 0 & 1 & 1 & 3 \\
0 & 0 & 0 & 1 & 4 \\
0 & 0 & 0 & 0 & 5
\end{pmatrix} \cdot (-1)^2
$$
 [/INST] Yes, we are allowed to interchange the columns of a matrix, but we must introduce a minus sign upon each swapping of a pair of columns. So, your calculation above is correct. The rules of row operations are the same as the rules of column operations because, as @Paul says in the comments above, $\det(A) = \det(A^T)$.
</s>