Changing sign when interchanging rows of matrix

Question

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

$$\det A = \begin{vmatrix} 2 & -1 & 1 & -1\\ 3 & 3 & 0 & 2\\ 1 & 2 & -1 & 1\\ 2 & 5 & 1 & 2 \end{vmatrix} = - \begin{vmatrix} 1 & 2 & -1 & 1\\ 3 & 3 & 0 & 2\\ 2 & -1 & 1 & -1\\ 2 & 5 & 1 & 2 \end{vmatrix} = - \begin{vmatrix} 1 & 2 & -1 & 1\\ 0 & -3 & 3 & -1\\ 0 & -5 & 3 & -3\\ 0 & 1 & 3 & 0 \end{vmatrix}$$

where, firstly, the 1st and 3rd row were swapped, changing sign of determinant, and then the 2nd and 4th rows were swapped (changing the sign of determinant or not?).

keep changing. determinant multiply by $-1$ for each switching of 2 rows (or columns). — Lance, Apr 29 '18 at 13:05
It would have to change each time. Imagine the other scenario. How could you decide whether someone had switched a pair of rows before they gave you the matrix in the first place? — Ethan Bolker, Apr 29 '18 at 13:12
Does this answer your question? Row swap changing sign of determinant — Rodrigo de Azevedo, Jun 15 '23 at 05:19

score 1 · Answer 1 · answered Apr 29 '18 at 13:56

1

The sign changes every time you switch rows.

answered Apr 29 '18 at 13:56

CyclotomicField

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Changing sign when interchanging rows of matrix

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