Hey, hopefully this is the correct place to post! I was looking for help with this textbook problem...

Find det D if...

Where the back of the book gives the answer as (-1)^{N}d_{1}d_{2}...d_{n}, N = n(n-1)/2.

This problem was actually asked before (Determinant Question), and I read the answer that was given and it made some sense, but still left me with questions. For example, can you even have a diagonal matrix from a 5x4 matrix? 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!