Question is as in title:

Why interchanging two rows of matrix results in negative determinant?

After thinking it a bit I felt:

interchanging rows 
= flipping orientation of transformation represented by matrix 
  change orientation of space
= -ve determinant / scaling factor of transformation 

Q1. Is it correct?
Q2. Also is this correct for all dimensions?
Q3. How flipping orientation of vector results in 3D?
Q3. Can we say interchanging rows result in change of basis? I feel basis are still same after interchanging rows right? First row still represents x-axis despite whether we interchange values in first row with 2 nd row (y axis) or 3 rd row (z axis) etc. Does changing orientation of space is a kind of change of basis?

  • 1
    $\begingroup$ If you use the permutation definition of a determinant, swapping two rows or columns amounts to swapping two elements of each permutation in the sum, which reverses the polarity of each permutation, reversing the sign of each element of the sum. $\endgroup$ – John Wayland Bales Jun 18 '20 at 23:22
  • $\begingroup$ Speaking of vectors and they order, then remember that changing the order in a cross product changes its sign. From here to a determinant... $\endgroup$ – Ripi2 Jun 18 '20 at 23:36
  • $\begingroup$ Another way of looking at it is by using $\det(A \cdot B) = \det (A) \det(B)$ and then convincing yourself that the determinant of a row permutation matrix is -1. $\endgroup$ – dskeletov Jun 19 '20 at 0:31

The determinant is the only, up to scalar multiplication, antisymmetric n-ary multilinear form on $\mathbb R^n$. What does that mean? Well, a multilinear form is a function of several vectors $f(x_1...x_n)$ such that $f$ is linear in every one of its arguments. Antisymmetry is when exchanging two arguments of $f$ causes $f$ to flip signs. As it turns out, there is only one possible antisymmetric multilinear function of $n$ $n$-dimensional vectors, and that's the determinant. It just-so-happens to be applicable to square matrices, if you put in each column as a vector argument.

So, from one perspective, antisymmetry is part of the definition of the determinant, and the surprising part is that it has a geometric interpretation at all.


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