can someone clear something for me?
I am confused about one property of determinants which is:
interchanging two rows or columns of a determinant changes the sign of the determinant.
Does it mean that when I interchange rows of a determinant several times the sign keeps changing or it changes just once?
Example:
$ det A =
\begin{vmatrix}
2 & -1 & 1 & -1\\
3 & 3 & 0 & 2\\
1 & 2 & -1 & 1\\
2 & 5 & 1 & 2
\end{vmatrix} = $
swaping 1st and 3rd row, changing sign of determinant.
$
det A = –
\begin{vmatrix}
1 & 2 & -1 & 1\\
3 & 3 & 0 & 2\\
2 & -1 & 1 & -1\\
2 & 5 & 1 & 2
\end{vmatrix}
= –
\begin{vmatrix}
1 & 2 & -1 & 1\\
0 & -3 & 3 & -1\\
0 & -5 & 3 & -3\\
0 & 1 & 3 & 0
\end{vmatrix}
$
swaping 2nd and 4th row (changing the sign of determinant or not?).
Best Answer
The sign changes every time you switch rows.