Why are you asking about a "5x4 matrix"? This question is clearly about an "nxn matrix" so definitely about square matrices.

The first thing I would do is try some simple examples: with n= 2, this would be which, with one swap of rows gives

With n= 2, there was 2(1)/2= 1 swap required.

With n= 3, we have .

Swapping the first and third rows immediately gives

But we could also swap the first and second rows, to get then swap second and third rows, and, finally, swap first and third rows to get again: that is 3(2)/2= 3 swaps. In any case the point is that that those have the same parity so the same sign.

No matter what won't there always be a "gap" since there would always be a column of zeroes? Also in the book the number of "switches" is given as n(n-1)/2 which using the logic of swapping rows to create a diagonal doesn't make sense (like if n = 5 that would be 10 swaps!)

Before consulting the web I thought the answer would have to do with cofactors and using the formula where the -1 and M multiply to be the cofactor of element a. As you can probably tell I'm pretty lost so any help would be great!