When does Gaussian elimination solve exact 1-in-3 SAT?
Does this always work on larger matrices with 2*n rows and 2*n+1 columns where n is the number of variables? (I think it may need non-redundant (linearly independent?) rows.)


It works but since there are assignments other than -1 and 1, how will you use the row echelon form to make any conclusions about the 1-in-3 SAT instance? 

Two cases: 


Case 1: The 1-in-3SAT formula is unsatisfiable but the reduced row echelon matrix is satisfiable (because of the possible assignments other than -1/1) 
Case 2: Both are satisfiable but assigning a 1 and -1 may bring you back to Case 1 


Therefore you will have to try an exponential number of assignments in worst case. Assigning random -1 and 1 values will force you to assign undesirable values during the end of elimination and you will have to backtrack. 

If you find an algorithm to find "only the solutions with -1 and 1" or count such solutions or prove there are 0 then it is NP-complete and even co-NP complete.