Proving Determinant of NxN Matrix All Rows & Cols Sum to 0

[robforsub]

If given a n*n matrix with all rows and columns sum to 0, how do I argue that all its (n-1)*(n-1) minor have the same determinant up to a sign?
Since all rows and columns all sum to 0, then I know that any column is a linear combination of all others, so that the determinant of this n*n matrix must be zero, then since the determinant is calculated using minors, it seems to imply that all (n-1)*(n-1) minors must have the same determinant up to a sign, but how do I rigorously prove that?

 

[Tinyboss]

This is just a hunch, but cofactor expansion and induction are probably involved.