Exercise 12

The second condition in the definition, that row swaps change the sign of a determinant, is somewhat annoying. It means we have to keep track of the number of swaps, to compute how the sign alternates. Can we get rid of it? Can we replace it with the condition that row swaps leave the determinant unchanged? (If so then we would need new $1\!\times\!1$, $2\!\times\!2$, and $3\!\times\!3$ formulas, but that would be a minor matter.)