det(cA)=c^n*det(a) where n is the number of rows so as long as you have an odd number of rows and the scalar is -1, then det(-A)=-det(A)
For #1, I currently have that the two values are equal if A is either a matrix whose determinant is zero or a 1x1 matrix.
I don't really have any direction for the others. I see that the determinant of a square matrix whose columns add to zero is 0, but I'm not sure of why. Can it somehow be reduced to the zero matrix?
Edit: Okay, messed around with row reduction. Quickly found that all columns adding to zero forces at least one row to row-reduce to a zero row, forcing a determinant of zero.
2) If are the columns it's asking for what given that . Let me ask you this: what is the determinant of a matrix whose columns are linearly dependent equal? Moreover, how does that apply to this question?
3) This is pretty straightforward. I don't know how to give a decent hint that doesn't give it away. Merely note that and go from there.
4) If is an eigenvalue for note that ...so...