Question

Is the Eigenvector [1,-1] equivalent to [-1,1]?

Answer 1

that is in a sense one eigenvector, because -1*(1,-1) = (-1,1)

Answer 2

so when I solve for an eigenvector using the eigenvalue and get something like [1 1] in the first row, the second row being just a multiple of the first, my eigenvector could be [1,-1] or [-1,1]?

Answer 3

its just a matter of preference. any multiple of an eigenvector is also an eigenvector because:
\[Ax = \lambda x\iff cAx = c\lambda x \iff A(cx) = \lambda (cx)\]