This is from my text. My instructor listed this as one for us to think about to help us prep for the exam on Tuesday. I really am stuck though, which is scaring me.

Here it goes:

Consider the matrix

where the 's and 's are fixed and the 's may vary. Show that a set of all such that is a subspace of of dimension at least 2.

I know that the determinant is a linear transformation in each column, but I don't really know how that helps me. I asked a classmate who said she thought Rank-Nullity theorem might help.. but I am still clueless. Any and all explanation on how to prove this would be great. Thanks!