Is this a valid proof for proving that swapping two rows in a matrix changes the sign of the determinant?

Question

Using the permutation definition of the determinant is this a valid proof for proving that swapping two rows in a matrix changes the sign of the determinant?

Picture of proof

Confusion part

I don't understand why the "-" sign comes in front of the (-1) because isn't this (the added permutation) already changing the sign on the determinant $(-1)^{σ(i1, …, il, …, ik, …, in)}$ ?

Shouldn't it be either:

$(-1)^{σ(i1, …, il, …, ik, …, in)}$

or

$-(-1)^{σ(i1, …, ik, …, il, …, in)}$

Answer 1

Hint: It is simple to prove it for $A=I$ identity Matrix. Thus you can observe if you change Two rows of a matrix $A$ then the new matrix $B$ is equal to

$B=P\cdot A$

where $P$ is a permutation matrix. This matrix is exactly that matrix obtained by the identity $I$ when you change two its rows. Thus you get

$det(B)=det(P)det(A)=-det(I)det(A)=-det(A)$